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A242817 a(n) = B(n,n), where B(n,x) = Sum_{k=0..n} Stirling2(n,k)*x^k are the Stirling polynomials. 13
1, 1, 6, 57, 756, 12880, 268098, 6593839, 187104200, 6016681467, 216229931110, 8588688990640, 373625770888956, 17666550789597073, 902162954264563306, 49482106424507339565, 2901159958960121863952, 181069240855214001514460, 11985869691525854175222222 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..370

FORMULA

E.g.f.: x*f'(x)/f(x), where f(x) is the generating series for sequence A035051.

a(n) ~ (exp(1/LambertW(1)-2)/LambertW(1))^n * n^n / sqrt(1+LambertW(1)). - Vaclav Kotesovec, May 23 2014

Conjecture: It appears that the equation a(x)*e^x = Sum_{n=0..oo} ( (n^x*x^n)/n! ) is true for every positive integer x. - Nicolas Nagel, Apr 20 2016

a(n) = n! * [x^n] exp(n*(exp(x)-1)). - Alois P. Heinz, May 17 2016

a(n) = [x^n] Sum_{k=0..n} n^k*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, May 31 2018

MAPLE

A:= proc(n, k) option remember; `if`(n=0, 1, (1+

      add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)

    end:

a:= n-> A(n$2):

seq(a(n), n=0..20);  # Alois P. Heinz, May 17 2016

MATHEMATICA

Table[BellB[n, n], {n, 0, 100}]

PROG

(Maxima) a(n):=stirling2(n, 0)+sum(stirling2(n, k)*n^k, k, 1, n);

makelist(a(n), n, 0, 30);

(PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*n^k); \\ Michel Marcus, Apr 20 2016

CROSSREFS

Cf. A035051, A292866.

Main diagonal of A189233 and of A292860.

Sequence in context: A305276 A032119 A294511 * A295238 A256016 A145170

Adjacent sequences:  A242814 A242815 A242816 * A242818 A242819 A242820

KEYWORD

nonn

AUTHOR

Emanuele Munarini, May 23 2014

STATUS

approved

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Last modified July 15 23:32 EDT 2020. Contains 335774 sequences. (Running on oeis4.)