A304193 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^2) / A(x) = 0 for n>0.

1, 4, 16, 144, 2346, 55236, 1688084, 63040736, 2770165274, 139623836116, 7925496107656, 499719554537584, 34625595715906866, 2613946666882042164, 213475621178226876156, 18748792440158256161216, 1761875767691411063734514, 176383456081424163875684516, 18739798321516251204837796864, 2105891800817103192582808107856

Offset: 0

Author

Paul D. Hanna, May 07 2018

Keywords

• nonn

Comments

Note that: [x^n] (1+x)^((n+1)*k) / G(x) = 0 for n>0 holds when G(x) = (1+x)^(k+1)/(1 - (k-1)*x) given some fixed k ; this sequence explores the case where k varies with n.

Examples

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 144*x^3 + 2346*x^4 + 55236*x^5 + 1688084*x^6 + 63040736*x^7 + 2770165274*x^8 + 139623836116*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^((n+1)^2) / A(x) begins:
n=0: [1, -3, -4, -80, -1530, -40222, -1316104, -51439572, ...];
n=1: [1, 0, -10, -100, -1785, -45056, -1441440, -55510080, ...];
n=2: [1, 5, 0, -140, -2380, -55080, -1685620, -63186200, ...];
n=3: [1, 12, 56, 0, -3150, -74484, -2125948, -76230384, ...];
n=4: [1, 21, 200, 1020, 0, -96492, -2901052, -98301840, ...];
n=5: [1, 32, 486, 4540, 26015, 0, -3718000, -135081440, ...];
n=6: [1, 45, 980, 13640, 132810, 855478, 0, -172046940, ...];
n=7: [1, 60, 1760, 33520, 462150, 4790156, 34461260, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n+1)^2) / A(x) = 0 for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 5, 56, 1020, 26015, 855478, 34461260, 1642995124, ...]
yields A132618, column 2 of triangle A132615.
Related triangular matrix T = A132615 begins:
1;
1, 1;
1, 1, 1;
6, 3, 1, 1;
80, 25, 5, 1, 1;
1666, 378, 56, 7, 1, 1;
47232, 8460, 1020, 99, 9, 1, 1;
1694704, 252087, 26015, 2134, 154, 11, 1, 1;
73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
in which row n equals row (n-1) of T^(2*n-1) followed by '1' for n > 0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A132618, A304190, A304191, A304192, A132615.

Programs

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^2)/Ser(A) )[m] ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

A132618(n+1) = [x^n] (1+x)^((n+2)^2) / A(x) for n>=0.