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%I #20 Jul 10 2022 19:16:41
%S 1,1,3,15,106,975,11100,151148,2401365,43681578,896371205,20504034645,
%T 517705752096,14310162565395,430020328711305,13963933247986995,
%U 487456219774434795,18209055555140970945,724952705958984299025,30650849492427960893946,1371796147488157950190065
%N Column 2 of triangle A107876.
%H Alois P. Heinz, <a href="/A107878/b107878.txt">Table of n, a(n) for n = 0..390</a>
%F G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^((k+1)*(k+2)/2).
%F From _Benedict W. J. Irwin_, Nov 29 2016: (Start)
%F Conjecture: a(n) is given by a series of nested sums,
%F a(1) = Sum_{i=1..1} 1,
%F a(2) = Sum_{i=1..1} Sum_{j=1..i+2} 1,
%F a(3) = Sum_{i=1..1} Sum_{j=1..i+2} Sum_{k=1..j+3} 1,
%F a(4) = Sum_{i=1..1} Sum_{j=1..i+2} Sum_{k=1..j+3} Sum_{l=1..k+4} 1. (End)
%e G.f. = 1 + x + 3*x^2 + 15*x^3 + 106*x^4 + 975*x^5 + 11100*x^6 + 151148*x^7 + ...
%e 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 +...
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(j-1)*
%p (-1)^(n-j)*binomial(j*(j+1)/2, n-j+1), j=1..n))
%p end:
%p seq(a(n), n=0..22); # _Alois P. Heinz_, Jul 10 2022
%t 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 *)
%o (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)}
%Y Cf. A000217, A107876, A107877, A107879, A209440 (variant).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 04 2005
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