cp's OEIS Frontend

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

%I A304193 #13 May 08 2018 23:24:35
%S A304193 1,4,16,144,2346,55236,1688084,63040736,2770165274,139623836116,
%T A304193 7925496107656,499719554537584,34625595715906866,2613946666882042164,
%U A304193 213475621178226876156,18748792440158256161216,1761875767691411063734514,176383456081424163875684516,18739798321516251204837796864,2105891800817103192582808107856
%N A304193 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^2) / A(x) = 0 for n>0.
%C A304193 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 A304193 Paul D. Hanna, <a href="/A304193/b304193.txt">Table of n, a(n) for n = 0..300</a>
%F A304193 A132618(n+1) = [x^n] (1+x)^((n+2)^2) / A(x) for n>=0.
%e A304193 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 A304193 ILLUSTRATION OF DEFINITION.
%e A304193 The table of coefficients of x^k in (1+x)^((n+1)^2) / A(x) begins:
%e A304193 n=0: [1, -3, -4, -80, -1530, -40222, -1316104, -51439572, ...];
%e A304193 n=1: [1, 0, -10, -100, -1785, -45056, -1441440, -55510080, ...];
%e A304193 n=2: [1, 5, 0, -140, -2380, -55080, -1685620, -63186200, ...];
%e A304193 n=3: [1, 12, 56, 0, -3150, -74484, -2125948, -76230384, ...];
%e A304193 n=4: [1, 21, 200, 1020, 0, -96492, -2901052, -98301840, ...];
%e A304193 n=5: [1, 32, 486, 4540, 26015, 0, -3718000, -135081440, ...];
%e A304193 n=6: [1, 45, 980, 13640, 132810, 855478, 0, -172046940, ...];
%e A304193 n=7: [1, 60, 1760, 33520, 462150, 4790156, 34461260, 0, ...]; ...
%e A304193 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 A304193 RELATED SEQUENCES.
%e A304193 The secondary diagonal in the above table that begins
%e A304193 [1, 5, 56, 1020, 26015, 855478, 34461260, 1642995124, ...]
%e A304193 yields A132618, column 2 of triangle A132615.
%e A304193 Related triangular matrix T = A132615 begins:
%e A304193 1;
%e A304193 1, 1;
%e A304193 1, 1, 1;
%e A304193 6, 3, 1, 1;
%e A304193 80, 25, 5, 1, 1;
%e A304193 1666, 378, 56, 7, 1, 1;
%e A304193 47232, 8460, 1020, 99, 9, 1, 1;
%e A304193 1694704, 252087, 26015, 2134, 154, 11, 1, 1;
%e A304193 73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
%e A304193 in which row n equals row (n-1) of T^(2*n-1) followed by '1' for n > 0.
%o A304193 (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 A304193 for(n=0,30, print1(a(n),", "))
%Y A304193 Cf. A132618, A304190, A304191, A304192, A132615.
%K A304193 nonn
%O A304193 0,2
%A A304193 _Paul D. Hanna_, May 07 2018