[seqfan] Re: A class of partitions.

[Robert G. Wilson v]

Et al,

	Using: f[n_] := Coefficient[ Series[ Product[1/(1 - q^k), {k, 1,
n}], {q, 0, 2 n + 3}], q^(2 n + 1)]; Array[f, 250] resulted in:

{1, 3, 8, 18, 37, 71, 131, 230, 393, 653, 1060, 1686, 2637, 4057, 6158,
9228, 13671, 20040, 29098, 41869, 59755, 84626, 118991, 166187, 230647,
318199, 436534, 595694, 808795, 1092876, 1470028, 1968738, 2625726, 3488072,
4616049, 6086556, 7997494, 10473078, 13670680, 17789019, 23078578, 29854292,
38511347, 49544424, 63571724, 81363984, 103880472, 132312710, 168138130,
213185155, 269713036, 340508206, 429001955, 539412651, 676918338, 847864940,
1060018609, 1322869014, 1647995783, 2049508702, 2544577387, 3154066342,
3903297786, 4822963731, 5950218807, 7329985295, 9016512170, 11075233375,
13584983909, 16640635254, 20356231598, 24868714067, 30342342347,
36973936069, 44999087187, 54699509838, 66411735748, 80537386256,
97555301392, 118035844788, 142657767964, 172228066034, 207705347412,
250227308765, 301143018341, 362050813226, 434842763488, 521756788744,
625437714744, 749008739232, 896155032619, 1071221455622, 1279326713376,
1526496601013, 1819819450621, 2167627341134, 2579707217708, 3067546685129,
3644620008849, 4326720666079, 5132347814279, 6083155120187, 7204471718467,
8525906512987, 10082048770670, 11913279844845, 14066713158465,
16597282064248, 19568998171239, 23056406020197, 27146263868760,
31939484637146, 37553376146931, 44124225397951, 51810278207957,
60795172899326, 71291895265840, 83547331602649, 97847507711698,
114523614211591, 133958932855390, 156596794717536, 182949719705375,
213609907731474, 249261275952056, 290693263476917, 338816655945190,
394681717273267, 459498955794715, 534662896939249, 621779286000751,
722696202350619, 839539632397549, 974754122934665, 1131149221092570,
1311952502388499, 1520870096778443, 1762155744642706, 2040689553210617,
2362067780095118, 2732705147615734, 3159951390923876, 3652223968863175,
4219159120724772, 4871783739751752, 5622710858177674, 6486361904634179,
7479219306646178, 8620113476245419, 9930548739948582, 11435073364742603,
13161699495689782, 15142379569152948, 17413546606769769, 20016726743017322,
22999233403683955, 26414953750798175, 30325239355296011, 34799914572165322,
39918417791565017, 45771092649074627, 52460648420420603, 60103811232262576,
68833190419194587, 78799387385987097, 90173377729047526, 103149201179813126,
117946998199459022, 134816436837721455, 154040578825378835,
175940239871423510, 200878905853184303, 229268274109858399,
261574497464710624, 298325218018626745, 340117488288588807,
387626689043209407, 441616566351932966, 502950525081821429,
572604332516468032, 651680404141225651, 741423864151192053,
843240596144821269, 958717525031750457, 1089645399720037857,
1238044377982070787, 1406192750413387488, 1596659180000914654,
1812338877976031188, 2056494185845787950, 2332800088351945829,
2645395243219960831, 2998939181640543855, 3398676409239510091,
3850508221708566029, 4361073143232433366, 4937837000428462377,
5589193760869691512, 6324578394706149224, 7154593161856767832,
8091148887323632445, 9147622965131398383, 10339036029190940049,
11682249449170993093, 13196186053650251755, 14902076754047056981,
16823736044034637067, 18987869683570839943, 21424418247913509331,
24166940633996661617, 27253042073719833628, 30724851710877257729,
34629555361004546286, 39019989697242304122, 43955304796977461704,
49501702750713247404, 55733260884346275431, 62732849087570551319,
70593151784233446469, 79417806235835703832, 89322670148874773410,
100437232973400215236, 112906186848434545371, 126891174885669918267,
142572736403645208339, 160152470849725197176, 179855444497996166619,
201932866611088960288, 226665064628870732969, 254364791125373841459,
285380898788709789059, 320102423561592422369, 358963120370626570109,
402446500612589198381, 451091425801788308166, 505498317565004174348,
566336050555501218594, 634349601906642978685, 710368538626550484522,
795316432922433755964, 890221304918719059040, 996227202686921304499,
1114607041036515032847, 1246776833233668971819, 1394311463838431341147,
1558962166312238938827, 1742675886091132259815, 1947616728604512660963,
2176189712419222237956, 2431067070495667739526}

Sincerely, Bob.



-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of William
Keith
Sent: Sunday, April 28, 2013 12:01 PM
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: A class of partitions.

Wouter, Edson:

     This number can be quickly calculated by a package as the coefficient
of q^(2n+1) in the initial segment of the Euler product for the partition
function, \prod_{k=1}^n 1/(1-q^k).  The restriction that there must be at
least 3 parts is redundant since to partition 2n+1 with parts up to size n,
you must have at least three parts (being greedy, you must use at least
n+n+1).

To any desired length:

RhomTable = Table[0, {i, 1, 40}];
For[n = 1, n < 41, n++,
 RhomTable[[n]] =
   Coefficient[
    Series[Product[1/(1 - q^k), {k, 1, n}], {q, 0, 2 n + 3}],
    q^(2 n + 1)];
 ]
RhomTable

     I strongly doubt that there is any closed formula.  To build a somewhat
complicated recursion, I would keep track of the largest part j and call
these numbers p_r(n,j).  Then a desired partition of 2n+1 is the sum over j
from 1 to n, and each p_r(n,j) arises by appending 1 <= k <=
floor((2n+1)/j) parts of size j to the total number of previous such
partitions of 2n+1-kj with largest parts of any size, plus the desired
partitions with exactly 3 parts, of which there are a small number.  You
will note that this also obliges us to keep track of a parallel series for
the even numbers, so there is much room if I think little chance for
improvement here.

     I would, of course, be cheerfully congratulatory if proved wrong.

Cordially,
William Keith

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