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<P>Consider the sequence f_k formed by concatenating k copies of each positive
integer, e.g:</P>
<P>f_3 = (1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 ...)</P>
<P>and let g_k be the sequence of partial sums of f_k:</P>
<P>g_3 = (1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 ...)</P>
<P>Also, let s_k be the periodic sequence with repeating block (1^(k-1), 0,
(-1)^(k-1), 0):</P>
<P>s_3 = (1 1 0 -1 -1 0 1 1 0 -1 -1 0 1 1 0 ...)</P>
<P>(f_k, g_k and s_k are indexed starting at 1).</P>
<P>The partition numbers p = (1 1 2 3 5 7 11 15 22 ...) = A000041 obey</P>
<P>p(0) = 1<BR>p(n) = SUM(i with g_3(i) <= n; s_3(i) * p(n - g_3(i)) (n >=
1).</P>
<P>This generalizes to</P>
<P>p_k(0) = 1<BR>p_k(n) = SUM(i with g_k(i) <= n; s_k(i) * p_k(n - g_k(i)) (n
>= 1)</P>
<P>with the partition numbers being the special case p = p_3.</P>
<P>------------------------------------------------------------------------<BR>Computing
a few terms of p_k for small k gives</P>
<P>p_1 = (1 0 0 0 0 0 0 0 0 0 0 0 0 0 ...)<BR>p_2 = (1 1 1 1 0 -1 -2 -3 -3 -1 2
6 10 11 ...)<BR>p_3 = (1 1 2 3 5 7 11 15 22 30 42 56 77 101 ...)<BR>p_4 = (1 1 2
4 7 13 23 42 75 135 242 434 779 1396 ...)<BR>p_5 = (1 1 2 4 8 15 29 55 106 202
387 739 1414 2702 ...)<BR>p_6 = (1 1 2 4 8 16 31 61 119 234 458 898 1759 3447
...)<BR>p_7 = (1 1 2 4 8 16 32 63 125 247 490 970 1922 3806 ...)<BR>p_8 = (1 1 2
4 8 16 32 64 127 253 503 1002 1994 3970 ...)<BR>p_9 = (1 1 2 4 8 16 32 64 128
255 509 1015 2026 4042 ...)<BR>p_10 = (1 1 2 4 8 16 32 64 128 256 511 1021 2039
4074 ...)</P>
<P>p_1 = A000007 and p_3 = A000041 and the termwise limit p_inf = A011782 are
easy. I was hoping for a combinatorial interpretation of other sequences, but
alas, they are not in the OEIS or elsewhere.</P>
<P>------------------------------------------------------------------------<BR>p_2
is interesting. It has the recurrence</P>
<P>p_2(0) = 1<BR>p_2(n) = SUM(k with 1 <= k^2 <= n; (-1)^(k+1) *
p_2(n-k^2)) (n >= 1)</P>
<P>or more prettily</P>
<P>p_2(0) = 1<BR>p_2(n) = p_2(n-1) - p_2(n-4) + p_2(n-9) - p_2(n-16) + p_2(n-25)
- ...</P>
<P>where terms with negative index are dropped in the latter sum.</P>
<P>p_2 is the only one of the p_n having negative elements. Here are
more<BR>terms of p_2, formatted to emphasize the regularity of the sign changes:
</P>
<P>p_2 = (<BR>1 1 1 1 0<BR>-1 -2 -3 -3 -1<BR>2 6 10 11 8 0<BR>-14 -29 -39 -38
-18<BR>22 74 123 144 110 6<BR>-161 -352 -491 -484 -251<BR>235 896 1528 1825 1452
191<BR>-1892 -4317 -6164 -6243 -3488<BR>2482 10788 18957 23140 19085
3858<BR>-22025 -52833 -77224 -80198 -47899<BR>25330 129563 234774 292984 249938
66467<BR>-254632 -645419 -966200 -1028145 -651774<BR>244756 1550984 2903014
3703662 3262048 1057768<BR>-2919617 -7868439 -12071433 -13154408
-8798790<BR>2156064 18502260 35840006 46747073 42441704 16042331<BR>-33149312
-95721387 -150601090 -167980582 -117979248<BR>15610769 219874960 441758527
589147690 550604495 235523164<BR>-371916746 -1161853462 -1876169578 -2141169812
-1572640509<BR>52991977 2601849672 5435999338 7413974493 7123950308
3377603117<BR>...)</P>
<P>------------------------------------------------------------------------<BR>Let
P_k be the g.f. for p_k. I am fairly convinced that</P>
<P>P_k = SUM(j = 0 to inf; (-(x^k))^j * Q_j)</P>
<P>where the Q_j are the g.f.'s for sequences q_j with</P>
<P>q_0 = (1 1 2 4 8 16 32 64 128 256 512 ...) = A011782<BR>q_1 = (1 3 9 22 54
126 290 654 1458 3214 7026 ...)<BR>q_2 = (1 5 20 64 189 519 1367 3475 8611 20887
49807 ...)<BR>...</P>
<P>I am not sure of the significance of these sequences. q_1 and q_2 are<BR>not
in the OEIS. I have not looked at q_3 or
beyond.<BR></P></FONT></FONT></DIV></BODY></HTML>