A304192 G.f. A(x) satisfies: [x^n] (1+x)^(n*(n+1)) / A(x) = 0 for n>0.

1, 2, 7, 72, 1224, 29184, 892074, 33144288, 1445847756, 72291575784, 4070550314292, 254674699992768, 17518238545282080, 1313558965998605568, 106608039857256267192, 9309469431887521270848, 870250987085629018699728, 86703492688056304091302944, 9171254392641669833788501488, 1026466161170552167031522911104

Offset: 0

Author

Paul D. Hanna, May 07 2018

Keywords

• nonn

Comments

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.

Examples

			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 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^(n*(n+1)) / A(x) begins:
n=0: [1, -2, -3, -52, -955, -24246, -771113, -29428232, ...];
n=1: [1, 0, -6, -60, -1062, -26208, -820560, -30994704, ...];
n=2: [1, 4, 0, -80, -1337, -30840, -932010, -34438500, ...];
n=3: [1, 10, 39, 0, -1722, -39996, -1138680, -40521096, ...];
n=4: [1, 18, 147, 648, 0, -50832, -1503546, -50844384, ...];
n=5: [1, 28, 372, 3048, 15465, 0, -1898490, -67990260, ...];
n=6: [1, 40, 774, 9580, 83248, 483240, 0, -85539792, ...];
n=7: [1, 54, 1425, 24420, 303363, 2844270, 18685905, 0, ...]; ...
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.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 4, 39, 648, 15465, 483240, 18685905, 861282832, 46085893011, ...]
yields A132612, column 1 of triangle A132610.
Related triangular matrix T = A132610 begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
194, 39, 6, 1, 1;
4114, 648, 76, 8, 1, 1;
118042, 15465, 1510, 125, 10, 1, 1;
4274612, 483240, 41121, 2908, 186, 12, 1, 1;
186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...
in which  row n+1 of T = row n of matrix power T^(2*n) with appended '1' for n>=0.
		

Links

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

Crossrefs

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

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

Formula

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