A278879 a(n) = Sum_{k=0..n-1} (k+1)*binomial(n,k) * a(k) for n > 0 with a(0) = 1.

1, 1, 5, 52, 931, 25516, 992799, 52032702, 3533592843, 301810098928, 31663565386063, 4002675000842794, 600053009189628075, 105257275528784647932, 21358127184625653306447, 4963922750189468652517318, 1310048479903507430878396651, 389626915538187147603694911208, 129712358816242576287065101448751, 48047325215595869387366562525477954

Offset

0,3

Comments

Equals the leftmost column in the limit of a triangular matrix B to the n-th power as n increases, where B(m,k) = (k+1)*binomial(m,k) for m > k with zeros along the main diagonal except for B(0,0) = 1.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Formula

a(n) ~ c * n^(2*n + 3/2) / exp(2*n), where c = 4.57091094257826787312511057805555932113507180436658388555642557963... . - Vaclav Kotesovec, Dec 09 2016

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 52*x^3/3! + 931*x^4/4! + 25516*x^5/5! + 992799*x^6/6! + 52032702*x^7/7! + 3533592843*x^8/8! + 301810098928*x^9/9! + 31663565386063*x^10/10! + ...

Prog

(PARI) {a(n) = if(n==0, 1, sum(k=0, n-1, (k+1)*binomial(n, k) * a(k) ))}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(B=matrix(n+1, n+1, m, k, if(m==1&&k==1, 1, if(m>k, k*binomial(m-1, k-1))))); (B^(n+1))[n+1, 1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A274257 A336609 A303000 * A322195 A216464 A279644

Adjacent sequences: A278876 A278877 A278878 * A278880 A278881 A278882

Keyword

nonn

Author

Paul D. Hanna, Dec 08 2016

Status

approved