A304187 G.f. A(x) satisfies: 0 = [x^n] (1+x)^((n+2)*(n+3)/2) / A(x) for n>0.

1, 6, 21, 86, 606, 6756, 102316, 1931046, 43250376, 1114876536, 32394654066, 1045240099026, 37027935179016, 1427410628324856, 59449956448178106, 2659215814433980056, 127108810653344820456, 6464722863550156435146, 348541208165221134718986, 19854709880058367829287716, 1191556960000156185148449636

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

A101479(n+4,3) = [x^n] (1+x)^((n+3)*(n+4)/2) / A(x) for n>=0.

Example

G.f.: A(x) = 1 + 6*x + 21*x^2 + 86*x^3 + 606*x^4 + 6756*x^5 + 102316*x^6 + 1931046*x^7 + 43250376*x^8 + 1114876536*x^9 + 32394654066*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^((n+1)*(n+2)/2) / A(x) begins:
n=0: [1, -3, 0, -22, -216, -3180, -56540, -1186170, -28599870, ...];
n=1: [1, 0, -6, -30, -285, -3894, -66750, -1365546, -32331180, ...];
n=2: [1, 4, 0, -50, -440, -5238, -84162, -1657080, -38209725, ...];
n=3: [1, 9, 30, 0, -645, -7917, -115248, -2134920, -47391375, ...];
n=4: [1, 15, 99, 335, 0, -11046, -171920, -2957874, -62097600, ...];
n=5: [1, 22, 225, 1378, 4984, 0, -233730, -4379370, -86791905, ...];
n=6: [1, 30, 429, 3850, 23610, 92652, 0, -5860422, -127938780, ...];
n=7: [1, 39, 735, 8875, 76350, 483684, 2065146, 0, -169402725, ...];
n=8: [1, 49, 1170, 18100, 203065, 1743201, 11567124, 53636520, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] (1+x)^((n+2)*(n+3)/2) / A(x) for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 4, 30, 335, 4984, 92652, 2065146, 53636520, 1589752230, ...]
yields column 3 of triangle A101479.
Related triangular matrix T = A101479 begins:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
191, 70, 18, 4, 1, 1;
2646, 795, 170, 30, 5, 1, 1;
46737, 11961, 2220, 335, 45, 6, 1, 1;
1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
in which row n equals row (n-1) of T^(n-1) followed by '1' for n>0.

Prog

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

Crossrefs

Cf. A101481, A304184, A304185, A304186, A101479.

Sequence in context: A088556 A316105 A137966 * A005498 A002222 A290355

Adjacent sequences: A304184 A304185 A304186 * A304188 A304189 A304190

Keyword

nonn

Author

Paul D. Hanna, May 08 2018

Status

approved