%I #20 Sep 30 2016 22:56:11
%S 1,1,1,3,14,96,989,14264,293081,8291372,326486284,17606371379,
%T 1311003529532,133789640100606,18842361596022104,3651812223033372781,
%U 979595054829206809506,363619011980801177687068,187594865162514096249150130,134684579087971548803896902904,134956937109764143572996094860839,189135846049140695927044178145555683,371258683769470709816610430835777163052
%N G.f.: Sum_{n>=0} (1-x)^(n*(n+1)) * [ Sum_{k>=1} k^n * x^k ]^n.
%H Paul D. Hanna, <a href="/A276747/b276747.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: Sum_{n>=0} [ Sum_{k=1..n} A008292(n,k) * x^k ]^n, where A008292 are the Eulerian numbers.
%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 96*x^5 + 989*x^6 + 14264*x^7 + 293081*x^8 + 8291372*x^9 + 326486284*x^10 + 17606371379*x^11 +...
%e such that
%e A(x) = Sum_{n>=0} (1-x)^(n*(n+1)) * (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n.
%e Explicitly,
%e A(x) = 1 + (1-x)^2 * (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 +...) +
%e (1-x)^6 * (x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 +...)^2 +
%e (1-x)^12 * (x + 8*x^2 + 27*x^3 + 64*x^4 + 125*x^5 + 216*x^6 +...)^3 +
%e (1-x)^20 * (x + 16*x^2 + 81*x^3 + 256*x^4 + 625*x^5 + 1296*x^6 +...)^4 +
%e (1-x)^30 * (x + 32*x^2 + 243*x^3 + 1024*x^4 + 3125*x^5 + 7776*x^6 +...)^5 +
%e ...
%e The g.f. can be written using the Eulerian numbers like so:
%e A(x) = 1 + x + (x + x^2)^2 + (x + 4*x^2 + x^3)^3 + (x + 11*x^2 + 11*x^3 + x^4)^4 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6 + (x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7)^7 + (x + 247*x^2 + 4293*x^3 + 15619*x^4 + 15619*x^5 + 4293*x^6 + 247*x^7 + x^8)^8 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n +...
%o (PARI) {a(n) = my(A=1); A = sum(m=0,n+1, (1-x +x*O(x^n))^(m*(m+1)) * sum(k=1,n+1,k^m*x^k +x*O(x^n))^m );polcoeff(A,n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
%o {a(n) = my(A=1, Oxn=x*O(x^n)); A = sum(m=0, n+1, sum(k=1, m, A008292(m, k)*x^k +Oxn )^m ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A276743, A276746, A008292.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 30 2016