A304643 G.f. A(x) satisfies: [x^n] (1+x)^(n^3) / A(x) = 0 for n>0.

1, 1, 21, 2075, 427745, 150754575, 80775206341, 61079788584715, 61918201760905701, 81032697606275994779, 132999148265782603510745, 267549402517056738883934727, 647439631215495429552890390761, 1855591663455916911410267165824087, 6216559993885861267930628826256971069, 24072412148295906199113974687972130690707, 106699538321376193436754733217464490904934733

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Example

G.f.: A(x) = 1 + x + 21*x^2 + 2075*x^3 + 427745*x^4 + 150754575*x^5 + 80775206341*x^6 + 61079788584715*x^7 + 61918201760905701*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^(n^3)/A(x) begins:
n=0: [1, -1, -20, -2034, -423216, -149819400, -80452969380, ...];
n=1: [1, 0, -21, -2054, -425250, -150242616, -80602788780, ...];
n=2: [1, 7, 0, -2166, -440034, -153263214, -81663489960, ...];
n=3: [1, 26, 304, 0, -470529, -161955486, -84652727166, ...];
n=4: [1, 63, 1932, 36334, 0, -174849912, -90924716676, ...];
n=5: [1, 124, 7605, 305466, 8541159, 0, -98844355155, ...];
n=6: [1, 215, 22984, 1626786, 85217850, 3329937702, 0, ...];
n=7: [1, 342, 58290, 6599344, 557724906, 37306986588, 1944420120804, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^(n^3) / A(x) = 0 for n>0.
RELATED SERIES.
1 - 1/A(x) = x + 20*x^2 + 2034*x^3 + 423216*x^4 + 149819400*x^5 + 80452969380*x^6 + 60910650903564*x^7 + 61792107766345152*x^8 + ...
The logarithmic derivative of the g.f. A(x) begins
A'(x)/A(x) = 1 + 41*x + 6163*x^2 + 1701881*x^3 + 751428751*x^4 + 483682989449*x^5 + 426965933360359*x^6 + 494840882952869729*x^7 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^((m-1)^3)/Ser(A) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A304191, A304644, A304398.

Sequence in context: A221122 A265175 A300577 * A153848 A221771 A007593

Adjacent sequences: A304640 A304641 A304642 * A304644 A304645 A304646

Keyword

nonn

Author

Paul D. Hanna, May 15 2018

Status

approved