A304185 G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n+1)/2) / A(x) for n>0.

1, 1, 1, 6, 51, 609, 9284, 171779, 3729929, 92828134, 2602268335, 81082146565, 2778537934965, 103826098384590, 4200631499008965, 182912327481135887, 8528448938007615767, 423908532350099458532, 22375155669520993773517, 1249871928128324730985042, 73662719257076005954584046

Offset

0,4

Links

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

Formula

[x^n] (1+x)^((n+1)*(n+2)/2) / A(x) = A101482(n+1) = A101479(n+2,1) for n>=0.

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

Example

G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 51*x^4 + 609*x^5 + 9284*x^6 + 171779*x^7 + 3729929*x^8 + 92828134*x^9 + 2602268335*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^(n*(n+1)/2) / A(x) begins:
n=0: [1, -1, 0, -5, -40, -513, -8092, -153395, -3388500, ...];
n=1: [1, 0, -1, -5, -45, -553, -8605, -161487, -3541895, ...];
n=2: [1, 2, 0, -7, -56, -648, -9756, -179250, -3873474, ...];
n=3: [1, 5, 9, 0, -75, -837, -11875, -210518, -4441140, ...];
n=4: [1, 9, 35, 70, 0, -1096, -15664, -263340, -5357885, ...];
n=5: [1, 14, 90, 345, 795, 0, -20260, -352235, -6842115, ...];
n=6: [1, 20, 189, 1115, 4510, 11961, 0, -452166, -9245340, ...];
n=7: [1, 27, 350, 2893, 17019, 74282, 224504, 0, -11809259, ...];
n=8: [1, 35, 594, 6505, 51545, 312984, 1483340, 5051866, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] (1+x)^(n*(n+1)/2) / A(x) for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 2, 9, 70, 795, 11961, 224504, 5051866, 132523155, ...]
yields A101482, column 1 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*(m-1)/2)/Ser(A) )[m] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A101481, A304184, A304186, A304187, A101479.

Sequence in context: A346667 A246189 A293128 * A215003 A356925 A134525

Adjacent sequences: A304182 A304183 A304184 * A304186 A304187 A304188

Keyword

nonn

Author

Paul D. Hanna, May 08 2018

Status

approved