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%I #10 Sep 29 2016 21:38:24
%S 1,1,3,12,63,447,4286,56185,1008317,24917676,849963761,40142633815,
%T 2633061525012,240207555735097,30578843349537575,5434894746337720676,
%U 1352812180415380719387,471689727423751377883607,230943183470327388401886858,158839247095790148049487792081,153694547774391577758847456894905
%N G.f.: Sum_{n>=0} [ Sum_{k>=1} k^n * x^k ]^n.
%H Paul D. Hanna, <a href="/A276743/b276743.txt">Table of n, a(n) for n = 0..150</a>
%F G.f.: Sum_{n>=0} [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n*(n+1)), where A008292 are the Eulerian numbers.
%e G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 63*x^4 + 447*x^5 + 4286*x^6 + 56185*x^7 + 1008317*x^8 + 24917676*x^9 + 849963761*x^10 +...
%e such that
%e A(x) = Sum_{n>=0} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n.
%e Explicitly,
%e A(x) = 1 + (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 +...) +
%e (x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 +...)^2 +
%e (x + 8*x^2 + 27*x^3 + 64*x^4 + 125*x^5 + 216*x^6 +...)^3 +
%e (x + 16*x^2 + 81*x^3 + 256*x^4 + 625*x^5 + 1296*x^6 +...)^4 +
%e (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/(1-x)^2 + (x + x^2)^2/(1-x)^6 + (x + 4*x^2 + x^3)^3/(1-x)^12 + (x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^20 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^30 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^42 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n*(n+1)) +...
%o (PARI) {a(n) = my(A=1);
%o A = sum(m=0,n+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) /* Using Eulerian numbers A008292 */
%o {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/(1-x +Oxn)^(m+1) )^m ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A276744, A276750.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 29 2016
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