A276748 G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} k^(n^2) * x^(n*k) ] / n ), a power series in x with integer coefficients.

1, 1, 3, 6, 20, 31, 278, 337, 17412, 24798, 6772374, 6838020, 11484638201, 11505059694, 80455953355044, 80659880546429, 2306084675313241000, 2306326405122809872, 268657126294137376567236, 268664044710902946519968, 126765866019584067600135507174, 126766706181193131138562011916, 241678197716027150352300025709078423, 241678578014230878979840920532089312, 1858396158247302094721803368957703312268486, 1858396883282148773045801834086535278817434

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..150

Formula

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.

Example

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 +...
such that
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.
The logarithm of g.f. A(x) equals the series:
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.
This logarithmic series can be written using the Eulerian numbers like so:
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 +...

Prog

(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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{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)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A276749, A008292.

Sequence in context: A356912 A176993 A359963 * A339639 A081181 A062164

Adjacent sequences: A276745 A276746 A276747 * A276749 A276750 A276751

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2016

Status

approved