A156325 E.g.f.: A(x) = exp( Sum_{n>=1} n(n+1)/2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0)=1.
1, 1, 4, 34, 482, 10056, 286372, 10591372, 491169996, 27826318000, 1887581200256, 150885500428224, 14028718134958936, 1500672248541122944, 182987661921689610000, 25231215606822797450176, 3906382859414378995123088, 674864208557744341737593088, 129369340822722468679389762496
Offset: 0
Keywords
Examples
E.g.f: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 482*x^4/4! + 10056*x^5/5! +... log(A(x)) = x + 3*1*x^2/2! + 6*4*x^3/3! + 10*34*x^4/4! + 15*482*x^5/5! +... such that log(A(x)) = x*A(x) + x^2*A'(x)/2 = d/dx x^2*A(x)/2.
Programs
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Mathematica
terms = 19; A[] = 1; Do[A[x] = Exp[x*A[x]+x^2*A'[x]/2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)
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PARI
{a(n) = if(n==0,1,n!*polcoeff(exp(sum(k=1,n,k*(k+1)/2*a(k-1)*x^k/k!)+x*O(x^n)),n))} for(n=0,25,print1(a(n),", ")) -
PARI
{a(n) = if(n==0,1,sum(k=1,n,k*(k+1)/2*binomial(n-1,k-1)*a(k-1)*a(n-k)))} for(n=0,25,print1(a(n),", ")) -
PARI
{a(n) = my(A=1); for(i=1,n, A = exp(deriv(x^2*A/2 +x^2*O(x^n)))); n!*polcoeff(A,n)} for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Dec 17 2017 -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+j/2)*binomial(i-1, j)*v[j+1]*v[i-j])); v; \\ Seiichi Manyama, Jul 25 2025
Formula
a(n) = Sum_{k=1..n} k(k+1)/2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.
E.g.f. A(x) satisfies A(x) = exp( d/dx x^2*A(x)/2 ). - Paul D. Hanna, Dec 17 2017
a(n) ~ c * n!^2 * n^3 / 2^n, where c = 0.1298868467763756870740708... - Vaclav Kotesovec, Aug 05 2025