A276906 G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} k^n * x^(2*k-1) ]^n / n ), a power series in x with integer coefficients.
1, 1, 1, 3, 7, 18, 53, 188, 799, 4001, 24050, 179248, 1639637, 17764040, 227653634, 3550628492, 67513114323, 1519274903363, 40153164845377, 1278514703044023, 49536414234360980, 2279497269454146657, 122986833567853232448, 7942922462379370617039, 622994706862172074402587, 58218522316121110190816538, 6379893924028925326363565894
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 18*x^5 + 53*x^6 + 188*x^7 + 799*x^8 + 4001*x^9 + 24050*x^10 + 179248*x^11 + 1639637*x^12 +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^n*x^3 + 3^n*x^5 +...+ k^n*x^(2*k-1) +...)^n/n.
Explicitly,
log(A(x)) = x + x^2/2 + 7*x^3/3 + 17*x^4/4 + 56*x^5/5 + 199*x^6/6 + 890*x^7/7 + 4649*x^8/8 + 27817*x^9/9 + 195946*x^10/10 + 1684398*x^11/11 + 17397323*x^12/12 +...+ A276907(n)*x^n/n +...
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = x/(1-x^2)^2 + (x + x^3)^2/(1-x^2)^6/2 + (x + 4*x^3 + x^5)^3/(1-x^2)^12/3 + (x + 11*x^3 + 11*x^5 + x^7)^4/(1-x^2)^20/4 + (x + 26*x^3 + 66*x^5 + 26*x^7 + x^9)^5/(1-x^2)^30/5 + (x + 57*x^3 + 302*x^5 + 302*x^7 + 57*x^9 + x^11)^6/(1-x^2)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * x^(2*k-1) ]^n / (1-x^2)^(n*(n+1))/n +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..200
Programs
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PARI
{a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^m*x^(2*k-1) +x*O(x^n))^m/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, m+1, A008292(m, k)*x^(2*k-1)/(1-x^2 +Oxn)^(m+1) )^m / m ) ); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", "))