OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..390
FORMULA
G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^((k+1)*(k+2)/2).
From Benedict W. J. Irwin, Nov 29 2016: (Start)
Conjecture: a(n) is given by a series of nested sums,
a(1) = Sum_{i=1..1} 1,
a(2) = Sum_{i=1..1} Sum_{j=1..i+2} 1,
a(3) = Sum_{i=1..1} Sum_{j=1..i+2} Sum_{k=1..j+3} 1,
a(4) = Sum_{i=1..1} Sum_{j=1..i+2} Sum_{k=1..j+3} Sum_{l=1..k+4} 1. (End)
EXAMPLE
G.f. = 1 + x + 3*x^2 + 15*x^3 + 106*x^4 + 975*x^5 + 11100*x^6 + 151148*x^7 + ...
1 = 1*(1-x)^1 + 1*x*(1-x)^3 + 3*x^2*(1-x)^6 + 15*x^3*(1-x)^10 + 106*x^4*(1-x)^15 + 975*x^5*(1-x)^21 + 11100*x^6*(1-x)^21 +...
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(a(j-1)*
(-1)^(n-j)*binomial(j*(j+1)/2, n-j+1), j=1..n))
end:
seq(a(n), n=0..22); # Alois P. Heinz, Jul 10 2022
MATHEMATICA
a[ n_, k_: 0, j_: 1] := If[ n < 1, Boole[n >= 0], a[ n, k, j] = Sum[ a[ n - 1, i, j + 1], {i, k + j}]]; (* Michael Somos, Nov 26 2016 *)
PROG
(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^((k+1)*(k+2)/2)), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 04 2005
STATUS
approved