OFFSET
0,4
FORMULA
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, ...]
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 08 2018
STATUS
approved