A183128 G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k>=0} C(n+k-1,k)^(k^2+1)*x^k]*x^n/n ).
1, 1, 2, 5, 131, 527019, 384803612051, 118132908813157848449, 7963186263790446068194034181927844, 116876153524994349756813783078174425848129593196964577
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 131*x^4 + 527019*x^5 +... The logarithm of the g.f. begins: log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 503*x^4/4 + 2634426*x^5/5 + 2308818509412*x^6/6 + 826930358998475963946*x^7/7 +... and equals the sum of the series: log(A(x)) = (1 + 1*x + 1*x^2 + 1*x^3 + 1*x^4 + 1*x^5 +...)*x + (1 + 2^2*x + 3^5*x^2 + 4^10*x^3 + 5^17*x^4 + 6^26*x^5 +...)*x^2/2 + (1 + 3^2*x + 6^5*x^2 + 10^10*x^3 + 15^17*x^4 + 21^26*x^5 +...)*x^3/3 + (1 + 4^2*x + 10^5*x^2 + 20^10*x^3 + 35^17*x^4 + 56^26*x^5 +...)*x^4/4 + (1 + 5^2*x + 15^5*x^2 + 35^10*x^3 + 70^17*x^4 + 126^26*x^5 +...)*x^5/5 + (1 + 6^2*x + 21^5*x^2 + 56^10*x^3 + 126^17*x^4 + 252^26*x^5 +...)*x^6/6 +...
Programs
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PARI
{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(k^2+1)*x^k)*x^m/m)+x*O(x^n)),n)} -
PARI
{a(n)=if(n==0, 1, (1/n)*sum(k=1, n, a(n-k)*sum(j=0, k-1, k*binomial(k-1, j)^(j^2+1)/(k-j))))}
Formula
a(n) = (1/n)*Sum_{k=1..n} L(k)*a(n-k) for n>0 with a(0) = 1, where L(n) = Sum_{k=0..n-1} n*C(n-1,k)^(k^2+1)/(n-k).
Comments