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A263843 Reversion of g.f. for A162395 (squares with signs). 3
0, 1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 4092553752, 37714212564, 350658882768, 3285490743987, 30989950019532, 294031964658430, 2804331954047160, 26870823304476690, 258548658860327880, 2497104592420003980, 24199830095943069360, 235254163727798051070 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is a variant of A007297, which is the main entry, with many references to both versions.

From Peter Bala, Apr 07 2020: (Start)

Let A(x) = 1 + 4*x + 23*x^2 + ... denote the o.g.f. of this sequence taken with an offset of 0. The sequence b(n) := [x^n] A(x)^n for n >= 1 begins [4, 62, 1084, 19982, 379504, 7347410, 144168392, 2856907662, 57044977168, 1145905776312, 23131265652092, ...]. We conjecture that the congruences b(n*p^k) == b(n*p^(k-1)) ( mod p^(3*k) ) hold for prime p >= 3 and all positive integers n and k.

More generally, for a positive integer r and integer s, the sequence b(r,s;n) := [x^(r*n)] A(x)^(s*n) is conjectured to satisfy the same congruences. (End)

LINKS

Table of n, a(n) for n=0..23.

FORMULA

a(n) ~ sqrt(7 - 4*sqrt(3)) * 2^(n-1/2) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 11 2017

MAPLE

with(gfun); t1:=(x-x^2)/(1+x)^3; t2:=series(t1, x, 50); t3:=seriestoseries(t2, 'revogf'); seriestolist(%);

MATHEMATICA

CoefficientList[InverseSeries[Series[x*(1-x)/(1+x)^3, {x, 0, 30}], x], x] (* Vaclav Kotesovec, Nov 11 2017 *)

CROSSREFS

Cf. A162395.

A variant of A007297.

Sequence in context: A055723 A271469 A007297 * A326350 A198916 A182969

Adjacent sequences:  A263840 A263841 A263842 * A263844 A263845 A263846

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 05 2015

STATUS

approved

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Last modified May 22 11:06 EDT 2022. Contains 353949 sequences. (Running on oeis4.)