A276748 - OEIS

%I #10 Sep 17 2016 22:55:22

%S 1,1,3,6,20,31,278,337,17412,24798,6772374,6838020,11484638201,

%T 11505059694,80455953355044,80659880546429,2306084675313241000,

%U 2306326405122809872,268657126294137376567236,268664044710902946519968,126765866019584067600135507174,126766706181193131138562011916,241678197716027150352300025709078423,241678578014230878979840920532089312,1858396158247302094721803368957703312268486,1858396883282148773045801834086535278817434

%N G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} k^(n^2) * x^(n*k) ] / n ), a power series in x with integer coefficients.

%H Paul D. Hanna, <a href="/A276748/b276748.txt">Table of n, a(n) for n = 0..150</a>

%F G.f.: exp( Sum_{n>=1} [ Sum_{k=1..n^2} A008292(n^2,k) * x^(n*k) ] / (1 - x^n)^(n^2+1) / n ), where A008292 are the Eulerian numbers.

%e G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 20*x^4 + 31*x^5 + 278*x^6 + 337*x^7 + 17412*x^8 + 24798*x^9 + 6772374*x^10 + 6838020*x^11 + 11484638201*x^12 +...

%e such that

%e log(A(x)) = Sum_{n>=1} (x^n + 2^(n^2)*x^(2*n) + 3^(n^2)*x^(3*n) +...+ k^(n^2)*x^(k*n) +...)/n.

%e The logarithm of g.f. A(x) equals the series:

%e log(A(x)) = Sum_{n>=1} (x^n + 2^(n^2)*x^(2*n) + 3^(n^2)*x^(3*n) +...+ 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^2 + 11*x^4 + 11*x^6 + x^8)/(1-x^2)^5/2 + (x^3 + 502*x^6 + 14608*x^9 + 88234*x^12 + 156190*x^15 + 88234*x^18 + 14608*x^21 + 502*x^24 + x^27)/(1-x^3)^10/3 + (x^4 + 65519*x^8 + 41932745*x^12 + 3572085255*x^16 + 85383238549*x^20 + 782115518299*x^24 + 3207483178157*x^28 + 6382798925475*x^32 + 6382798925475*x^36 + 3207483178157*x^40 + 782115518299*x^44 + 85383238549*x^48 + 3572085255*x^52 + 41932745*x^56 + 65519*x^60 + x^64)/(1-x^4)^17/4 + (x^5 + 33554406*x^10 + 846416194536*x^15 + 1103881308184906*x^20 + 269025107855605626*x^25 + 21045399230106913746*x^30 + 695824003645512474376*x^35 + 11392907456028953400606*x^40 + 101955892318210543172751*x^45 + 531714261368950897339996*x^50 + 1685388700882132120106256*x^55 + 3334612565134607644610436*x^60 + 4179647109945703200884716*x^65 + 3334612565134607644610436*x^70 + 1685388700882132120106256*x^75 + 531714261368950897339996*x^80 + 101955892318210543172751*x^85 + 11392907456028953400606*x^90 + 695824003645512474376*x^95 + 21045399230106913746*x^100 + 269025107855605626*x^105 + 1103881308184906*x^110 + 846416194536*x^115 + 33554406*x^120 + x^125)/(1-x^5)^26/5 +...+ [Sum_{k=1..n^2} A008292(n^2,k) * x^(n*k)]/(1 - x^n)^(n^2+1)/n +...

%o (PARI) {a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n\m+1, k^(m^2) * x^(m*k) +x*O(x^n)) / m ) ), 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 = exp( sum(m=1, n+1, sum(k=1, min(m^2,n)+1, A008292(m^2, k)*x^(m*k)/(1-x^m +Oxn)^(m^2+1) ) / m ) ); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A276749, A008292.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 17 2016