A304192 - OEIS

%I #17 May 09 2018 00:33:26

%S 1,2,7,72,1224,29184,892074,33144288,1445847756,72291575784,

%T 4070550314292,254674699992768,17518238545282080,1313558965998605568,

%U 106608039857256267192,9309469431887521270848,870250987085629018699728,86703492688056304091302944,9171254392641669833788501488,1026466161170552167031522911104

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

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

%H Paul D. Hanna, <a href="/A304192/b304192.txt">Table of n, a(n) for n = 0..300</a>

%F A132612(n+1) = [x^n] (1+x)^((n+1)*(n+2)) / A(x) for n>0.

%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 72*x^3 + 1224*x^4 + 29184*x^5 + 892074*x^6 + 33144288*x^7 + 1445847756*x^8 + 72291575784*x^9 + ...

%e ILLUSTRATION OF DEFINITION.

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

%e n=0: [1, -2, -3, -52, -955, -24246, -771113, -29428232, ...];

%e n=1: [1, 0, -6, -60, -1062, -26208, -820560, -30994704, ...];

%e n=2: [1, 4, 0, -80, -1337, -30840, -932010, -34438500, ...];

%e n=3: [1, 10, 39, 0, -1722, -39996, -1138680, -40521096, ...];

%e n=4: [1, 18, 147, 648, 0, -50832, -1503546, -50844384, ...];

%e n=5: [1, 28, 372, 3048, 15465, 0, -1898490, -67990260, ...];

%e n=6: [1, 40, 774, 9580, 83248, 483240, 0, -85539792, ...];

%e n=7: [1, 54, 1425, 24420, 303363, 2844270, 18685905, 0, ...]; ...

%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^(n*(n+1)) / A(x) = 0 for n>0.

%e RELATED SEQUENCES.

%e The secondary diagonal in the above table that begins

%e [1, 4, 39, 648, 15465, 483240, 18685905, 861282832, 46085893011, ...]

%e yields A132612, column 1 of triangle A132610.

%e Related triangular matrix T = A132610 begins:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 14, 4, 1, 1;

%e 194, 39, 6, 1, 1;

%e 4114, 648, 76, 8, 1, 1;

%e 118042, 15465, 1510, 125, 10, 1, 1;

%e 4274612, 483240, 41121, 2908, 186, 12, 1, 1;

%e 186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...

%e in which  row n+1 of T = row n of matrix power T^(2*n) with appended '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*(m-1))/Ser(A) )[m] ); A[n+1]}

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

%Y Cf. A132612, A304189, A304188, A304191, A304193, A132610.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 07 2018