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A202476 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x + x^2).

%I A202476 #14 Nov 02 2014 12:32:39
%S A202476 1,1,2,5,11,28,74,206,601,1826,5766,18851,63676,221678,793958,2920292,
%T A202476 11014653,42543773,168074091,678403932,2794920078,11742254750,
%U A202476 50266213000,219085792538,971543475593,4380664101448,20071848941411,93403455862117,441206005123701
%N A202476 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x + x^2).
%H A202476 Vaclav Kotesovec, <a href="/A202476/b202476.txt">Table of n, a(n) for n = 0..300</a>
%F A202476 Antidiagonal sums of the irregular triangle in which row n is defined by the g.f.: Product_{k=1..n} (1 + k*x + x^2) for n>=0.
%F A202476 G.f.: 1/(1 - x*(1+x+x^2)/(1 + x*(1+x+x^2) - x*(1+2*x+x^2)/(1 + x*(1+2*x+x^2) - x*(1+3*x+x^2)/(1 + x*(1+3*x+x^2) - x*(1+4*x+x^2)/(1 + x*(1+4*x+x^2) -...))))), a continued fraction.
%e A202476 The coefficients in Product_{k=1..n} (1+k*x+x^2), n>=0, form the triangle:
%e A202476 [1];
%e A202476 [1, 1, 1];
%e A202476 [1, 3, 4, 3, 1];
%e A202476 [1, 6, 14, 18, 14, 6, 1];
%e A202476 [1, 10, 39, 80, 100, 80, 39, 10, 1];
%e A202476 [1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1];
%e A202476 [1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1];
%e A202476 [1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
%e A202476 the antidiagonal sums of which form this sequence.
%o A202476 (PARI) {a(n)=sum(k=0,n,polcoeff(prod(j=1,n-k,1+j*x+x^2),k))}
%o A202476 (PARI) {a(n)=local(CF=1+x+x*O(x^n)); for(k=1, n-1, CF=(1+(n-k)*x+x^2)/(1 + x*(1+(n-k)*x+x^2) - x*CF+x*O(x^n))); polcoeff(1/(1-x*CF), n)}
%Y A202476 Cf. A201951, A201826, A202474.
%K A202476 nonn
%O A202476 0,3
%A A202476 _Paul D. Hanna_, Dec 19 2011