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%I #5 Sep 17 2016 23:27:44
%S 1,1,3,22,749,349707,6584568222,2542670826073083,
%T 87482825374559636232439,1084004198573118046271860417698544,
%U 947790766920144318254338856937912501990845477,546110521982991331256716555878135043551458467258822092049841,1013482348116310649878997474896504367633097553028647215670516799670576593506574
%N G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^(n^2) * x^k]^n / n ), a power series in x with integer coefficients.
%H Paul D. Hanna, <a href="/A276749/b276749.txt">Table of n, a(n) for n = 0..30</a>
%F G.f.: exp( Sum_{n>=1} [ Sum_{k=1..n^2} A008292(n^2,k) * x^k / (1-x)^(n^2+1) ]^n / n ), where A008292 are the Eulerian numbers.
%e G.f.: A(x) = 1 + x + 3*x^2 + 22*x^3 + 749*x^4 + 349707*x^5 + 6584568222*x^6 + 2542670826073083*x^7 + 87482825374559636232439*x^8 +...
%e log(A(x)) = x + 5*x^2/2 + 58*x^3/3 + 2901*x^4/4 + 1744601*x^5/5 + 39505301300*x^6/6 + 17798649685552457*x^7/7 + 699862582655005078651885*x^8/8 + 9756037786370716942306622514588154*x^9/9 +...
%e The logarithm of g.f. A(x) equals the series:
%e log(A(x)) = Sum_{n>=1} (x + 2^(n^2)*x^2 + 3^(n^2)*x^3 +...+ k^(n^2)*x^k +...)^n/n.
%e This logarithmic series can be written using the Eulerian numbers like so:
%e log(A(x)) = x/(1-x)^2 + (x + 11*x^2 + 11*x^3 + x^4)^2/(1-x)^10/2 + (x + 502*x^2 + 14608*x^3 + 88234*x^4 + 156190*x^5 + 88234*x^6 + 14608*x^7 + 502*x^8 + x^9)^3/(1-x)^30/3 + (x + 65519*x^2 + 41932745*x^3 + 3572085255*x^4 + 85383238549*x^5 + 782115518299*x^6 + 3207483178157*x^7 + 6382798925475*x^8 + 6382798925475*x^9 + 3207483178157*x^10 + 782115518299*x^11 + 85383238549*x^12 + 3572085255*x^13 + 41932745*x^14 + 65519*x^15 + x^16)^4/(1-x)^68/4 + (x + 33554406*x^2 + 846416194536*x^3 + 1103881308184906*x^4 + 269025107855605626*x^5 + 21045399230106913746*x^6 + 695824003645512474376*x^7 + 11392907456028953400606*x^8 + 101955892318210543172751*x^9 + 531714261368950897339996*x^10 + 1685388700882132120106256*x^11 + 3334612565134607644610436*x^12 + 4179647109945703200884716*x^13 + 3334612565134607644610436*x^14 + 1685388700882132120106256*x^15 + 531714261368950897339996*x^16 + 101955892318210543172751*x^17 + 11392907456028953400606*x^18 + 695824003645512474376*x^19 + 21045399230106913746*x^20 + 269025107855605626*x^21 + 1103881308184906*x^22 + 846416194536*x^23 + 33554406*x^24 + x^25)^5/(1-x)^130/5 +...+ [Sum_{k=1..n^2} A008292(n^2,k) * x^k]^n / (1 - x^n)^(n^3+n) /n +...
%o (PARI) {a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^(m^2) * x^k +x*O(x^n))^m / m ) ), n)}
%o for(n=0, 15, 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 = exp( sum(m=1, n+1, sum(k=1, m^2, A008292(m^2, k)*x^k/(1-x +Oxn)^(m^2+1) )^m / m ) ); polcoeff(A, n)}
%o for(n=0, 15, print1(a(n), ", "))
%Y Cf. A156170, A276748, A008292.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 17 2016
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