A304193 - OEIS

%I #13 May 08 2018 23:24:35

%S 1,4,16,144,2346,55236,1688084,63040736,2770165274,139623836116,

%T 7925496107656,499719554537584,34625595715906866,2613946666882042164,

%U 213475621178226876156,18748792440158256161216,1761875767691411063734514,176383456081424163875684516,18739798321516251204837796864,2105891800817103192582808107856

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

%C 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.

%H Paul D. Hanna, <a href="/A304193/b304193.txt">Table of n, a(n) for n = 0..300</a>

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

%e 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 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in (1+x)^((n+1)^2) / A(x) begins:

%e n=0: [1, -3, -4, -80, -1530, -40222, -1316104, -51439572, ...];

%e n=1: [1, 0, -10, -100, -1785, -45056, -1441440, -55510080, ...];

%e n=2: [1, 5, 0, -140, -2380, -55080, -1685620, -63186200, ...];

%e n=3: [1, 12, 56, 0, -3150, -74484, -2125948, -76230384, ...];

%e n=4: [1, 21, 200, 1020, 0, -96492, -2901052, -98301840, ...];

%e n=5: [1, 32, 486, 4540, 26015, 0, -3718000, -135081440, ...];

%e n=6: [1, 45, 980, 13640, 132810, 855478, 0, -172046940, ...];

%e n=7: [1, 60, 1760, 33520, 462150, 4790156, 34461260, 0, ...]; ...

%e 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.

%e RELATED SEQUENCES.

%e The secondary diagonal in the above table that begins

%e [1, 5, 56, 1020, 26015, 855478, 34461260, 1642995124, ...]

%e yields A132618, column 2 of triangle A132615.

%e Related triangular matrix T = A132615 begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 6, 3, 1, 1;

%e 80, 25, 5, 1, 1;

%e 1666, 378, 56, 7, 1, 1;

%e 47232, 8460, 1020, 99, 9, 1, 1;

%e 1694704, 252087, 26015, 2134, 154, 11, 1, 1;

%e 73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...

%e in which row n equals row (n-1) of T^(2*n-1) followed by '1' for n > 0.

%o (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]}

%o for(n=0,30, print1(a(n),", "))

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 07 2018