Cleaning out the junk

[David Wilson]

I'm cleaning up junk from my sequence researches. Here is an old note I 
wrote to myself in case anyone is interested. I don't think I ever acted on 
it by noting or creating sequences.

When we compute a few terms of p_k for k = 1 through 10, we get

p_1 = (1 0 0 0 0 0 0 0 0 0 0 0 0 0 ...)

p_2 = (1 1 1 1 0 -1 -2 -3 -3 -1 2 6 10 11 ...)

p_3 = (1 1 2 3 5 7 11 15 22 30 42 56 77 101 ...)

p_4 = (1 1 2 4 7 13 23 42 75 135 242 434 779 1396 ...)

p_5 = (1 1 2 4 8 15 29 55 106 202 387 739 1414 2702 ...)

p_6 = (1 1 2 4 8 16 31 61 119 234 458 898 1759 3447 ...)

p_7 = (1 1 2 4 8 16 32 63 125 247 490 970 1922 3806 ...)

p_8 = (1 1 2 4 8 16 32 64 127 253 503 1002 1994 3970 ...)

p_9 = (1 1 2 4 8 16 32 64 128 255 509 1015 2026 4042 ...)

p_10 = (1 1 2 4 8 16 32 64 128 256 511 1021 2039 4074 ...)

p_1 = A000007 and p_3 = A000041 and the termwise limit p_inf = A011782 are 
straightforward. None of the other sequences appear to be in the OEIS.

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p_2 is interesting. It has the recurrence

p_2(0) = 1

p_2(n) = SUM(k with 1 <= k^2 <= n; (-1)^(k+1) * p_2(n-k^2)) (n >= 1)

or more prettily

p_2(0) = 1

p_2(n) = p_2(n-1) - p_2(n-4) + p_2(n-9) - p_2(n-16) + p_2(n-25) - ...

where terms with negative index are dropped in the latter sum.

p_2 is the only one of the p_n having negative elements. Here I provide more 
terms, formatted to emphasize the regularity of the sign changes:

p_2 = (

1 1 1 1 0

-1 -2 -3 -3 -1

2 6 10 11 8 0

-14 -29 -39 -38 -18

22 74 123 144 110 6

-161 -352 -491 -484 -251

235 896 1528 1825 1452 191

-1892 -4317 -6164 -6243 -3488

2482 10788 18957 23140 19085 3858

-22025 -52833 -77224 -80198 -47899

25330 129563 234774 292984 249938 66467

-254632 -645419 -966200 -1028145 -651774

244756 1550984 2903014 3703662 3262048 1057768

-2919617 -7868439 -12071433 -13154408 -8798790

2156064 18502260 35840006 46747073 42441704 16042331

-33149312 -95721387 -150601090 -167980582 -117979248

15610769 219874960 441758527 589147690 550604495 235523164

-371916746 -1161853462 -1876169578 -2141169812 -1572640509

52991977 2601849672 5435999338 7413974493 7123950308 3377603117

...)

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Let P_k be the g.f. for p_k. I am fairly convinced that

P_k = SUM(j = 0 to inf; (-(x^k))^j * Q_j)

where the Q_j is the g.f. for q_j with

q_0 = (1 1 2 4 8 16 32 64 128 256 512 ...) = A011782

q_1 = (1 3 9 22 54 126 290 654 1458 3214 7026 ...)

q_2 = (1 5 20 64 189 519 1367 3475 8611 20887 49807 ...)

...

q_1 and q_2 are not in the OEIS. I have not looked at q_3 or beyond.