A304184 G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n-1)/2) / A(x) for n>0.

1, 0, 0, 1, 9, 117, 1851, 34923, 765933, 19155084, 538051164, 16771165230, 574424285076, 21443516818065, 866521903003641, 37683366660458208, 1754777541925339779, 87115221430910051901, 4592968693335470802627, 256294382115032521083411, 15090698035153332532531074

Offset

0,5

Links

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

Formula

[x^n] (1+x)^(n*(n+1)/2) / A(x) = A101481(n+1) = A101479(n+1,0) for n>=0.

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

Example

G.f.: A(x) = 1 + x^3 + 9*x^4 + 117*x^5 + 1851*x^6 + 34923*x^7 + 765933*x^8 + 19155084*x^9 + 538051164*x^10 + 16771165230*x^11 + 574424285076*x^12 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^(n*(n-1)/2) / A(x) begins:
n=0: [1, 0, 0, -1, -9, -117, -1850, -34905, -765618, ...];
n=1: [1, 0, 0, -1, -9, -117, -1850, -34905, -765618, ...];
n=2: [1, 1, 0, -1, -10, -126, -1967, -36755, -800523, ...];
n=3: [1, 3, 3, 0, -12, -147, -2229, -40815, -876000, ...];
n=4: [1, 6, 15, 19, 0, -180, -2706, -47955, -1005279, ...];
n=5: [1, 10, 45, 119, 191, 0, -3335, -59840, -1214055, ...];
n=6: [1, 15, 105, 454, 1341, 2646, 0, -73965, -1545531, ...];
n=7: [1, 21, 210, 1329, 5955, 19833, 46737, 0, -1913457, ...];
n=8: [1, 28, 378, 3275, 20438, 97533, 364936, 1003150, 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, 1, 3, 19, 191, 2646, 46737, 1003150, 25330125,  ...]
yields A101481, column 0 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, A304185, A304186, A304187, A304189, A101479.

Sequence in context: A062994 A059967 A346769 * A255521 A027396 A294190

Adjacent sequences: A304181 A304182 A304183 * A304185 A304186 A304187

Keyword

nonn

Author

Paul D. Hanna, May 08 2018

Status

approved