A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - n*k*x)).

1, 1, 6, 105, 3568, 204745, 18028266, 2278860535, 394667414016, 90302033890953, 26525942216131330, 9775058594870836861, 4433256936788979640848, 2434899483389881601250937, 1597444746833206096334387802, 1237091666097626095124512681755, 1119205949224015886848972396596736

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..212

Formula

Sum_{n>=0} a(n)*x^n / n!^2 = Sum_{n>=0} (exp(n*x) - 1)^n / n!^2.

a(n) = n! * Sum_{k=0..n} k^n * Stirling2(n,k) / k!.

Example

E.g.f.: A(x) = 1 + x + 6*x^2/2! + 105*x^3/3! + 3568*x^4/4! + 204745*x^5/5! +...
where
A(x) = 1 + x/(1-x) + 2^2*x^2/(2!*(1-2*1*x)*(1-2*2*x)) + 3^3*x^3/(3!*(1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4^4*x^4/(4!*(1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...

Mathematica

Flatten[{1, Table[n! * Sum[k^n * StirlingS2[n, k] / k!, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 08 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(sum(m=0, 20, m^m*x^m/m!/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!^2*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!^2), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!*sum(k=0, n, k^n*stirling(n, k, 2)/k!)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A119392, A229257, A229258, A229259, A229260, A229261.

Cf. A229233, A229234, A220181, A122399.

Sequence in context: A309378 A221933 A110342 * A126467 A013294 A013300

Adjacent sequences: A227204 A227205 A227206 * A227208 A227209 A227210

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2013

Status

approved