chess tournaments:

[David W. Wilson]

Svante Linusson wrote:

> >We can parameterize these sequences on the total number of pointsp awarded
> >per game.  In such a game, there are p+1
> >possible game
> >outcomes, namely (0,p), (1,p-1), ..., and (p,0).  Let A_p(n) count
> >the number of score sequences for n players for n = 1, 2, 3, ....
> >
> >The A047657 = A_2, and your other requested sequence is A_3.
> >I have computed the first few values for A_p, for 1 <= p <= 5.
> >
> >p       A_p
> >
> >1       1,1,2,4,9,22,59,167,490,1486
> >2       1,2,5,16,59,247,1111,5302,26376
> >3       1,2,8,37,198,1178,7548
> >4       1,3,13,76,521,3996,32923
> >5       1,3,18,131,1111,10461
> >
> >Notes:
> >
> >    A047657 should be indexed starting at 1.
> >    A_1 is probably the same as A000571.  Whereas A_1 has a recurrence,
> >        it might be possible to work one out for all A_p.
> >    A_2 is probably the same as A007747.
>
> By changing <=2 in the description of A007747 to <=p, you should get the
> same sequences.
>
> /Svante
>
> =========================================================
> Svante Linusson
> Matematiska Institutionen       tel. 08-16 14 34
> Stockholms Universitet          fax. 08-612 67 17
> S-10691 Stockholm               linusson at matematik.su.se
> SWEDEN


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I independently arrived at this conclusion, and when I received
Svante's note, I was already computing variants of A007747, replacing
"<= 2" in the definition with "<= p" for p = 1 through 10.  On the
basis of the empirical results, I agree with Svante that changing
"<= 2" to "<= p" in the defintion of A007747 yields the score counting
sequence with p points awarded per game.

Below find my computations of the A007747 variants with 1 <= p <= 10.
These coincide with my earlier computations of the score counting
sequences with p points awarded per game (quoted above), as far as the
latter go.  Each sequence is indexed starting at n = 1.

p = 1
1,1,2,4,9,22,59,167,490,1486,4639,14805,48107,158808,531469,1799659,
6157068,21258104,73996100,259451116,915695102,3251073303,11605141649,
41631194766,150021775417,542875459724,1972050156181,7189259574618
Note: Apparently extends A000571.

p = 2
1,2,5,16,59,247,1111,5302,26376,135670,716542,3868142,21265884,
118741369,671906876,3846342253,22243294360,129793088770,763444949789,
4522896682789,26968749517543,161750625450884,975311942386969
Note: Extends A007747, apparently extends A047657.

p = 3
1,2,8,37,198,1178,7548,50944,357855,2595250,19313372,146815503,
1136158495,8927025989,71065654235,572215412354,4653746621835,
38184724333615,315792633485360,2630183440412617,22046522161472304

p = 4:
1,3,13,76,521,3996,32923,286202,2590347,24203935,232050202,2272449745,
22653570386,229274897514,2350933487206,24381053759852,255382755251622,
2698732882975782,28743579211912338,308306154951518523,3328001652757405282

p = 5:
1,3,18,131,1111,10461,105819,1127413,12499673,143021541,1678718575,
20123155604,245521479531,3041006378312,38157059717410,484209044329613,
6205758830280388,80235572611152385,1045527493257497732,13719888598988064954

p = 6:
1,4,25,213,2131,23729,283681,3574222,46866712,634204317,8803501719,
124799484286,1800669899917,26374204955323,391331674556361,
5872226011836383,88993282402441857,1360552594176453319

p = 7:
1,4,32,318,3692,47536,657040,9563961,144847330,2263567060,36281911266,
593856894136,9892591942306,167278802007062,2865331941321996,
49634901816988932,868329574365547207,15324138148802269892

p = 8:
1,5,41,459,6033,88055,1379405,22763356,390859501,6924877318,
125837754305,2335060741480,44097660919285,845336236860344,
16415016380975679,322349248087651458,6392828942756895663

p = 9:
1,5,50,630,9285,151652,2658131,49061128,942055396,18662965393,
379195887105,7867076520341,166102773740621,3559787677138284,
77278541685154409,1696519572528877274

p = 10:
1,6,61,846,13771,248623,4816659,98277943,2086173336,45688601782,
1026218795502,23536101285148,549336702455778,13014352354398322,
312313455482385108,7579157833713922471