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A015735 Row sums of triangle A004747. 6
1, 3, 17, 145, 1661, 23931, 415773, 8460257, 197360985, 5192853011, 152137882601, 4911873672113, 173268075672277, 6630323916472075, 273555262963272501, 12105084133976359361, 571897644855277242673 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

FORMULA

E.g.f.: exp(1-(1-3*x)^(1/3))-1, if one takes a(0)=0.

a(n) = 6*(n-2)*a(n-1) - (3*n-8)*(3*n-7)*a(n-2) + a(n-3), a(0)=1, a(1)=1, a(2)=3.

a(n) = 1 + (n-1)!*Sum_{m=1..n} (Sum_{k=1..n-m} C(k, n-m-k)*(-1/3)^(n-m-k)*C(k+n-1, n-1)) / (m-1)!, n > 1. - Vladimir Kruchinin, Aug 08 2010

a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^2*d/dx. Cf. A001515, A016036 and A028575. - Peter Bala, Nov 25 2011

E.g.f. with offset 0: exp(1-(1-3*x)^(1/3))/(1-3*x)^(2/3). - Sergei N. Gladkovskii, Jul 07 2012.

a(n) ~ sqrt(2*Pi)*3^(n-1)*exp(1-n)*n^(n-5/6)/GAMMA(2/3) * (1-sqrt(3)*GAMMA(2/3)^2/(2*Pi*n^(1/3))). - Vaclav Kotesovec, Aug 10 2013

Recurrence: a(n) = 6*(n-2)*a(n-1) - (3*n-8)*(3*n-7)*a(n-2) + a(n-3). - Vaclav Kotesovec, Aug 10 2013

MATHEMATICA

a[1] = 1; a[n_] := (n-1)!*Sum[ Sum[ Binomial[k, n-m-k]*(-1/3)^(n-m-k)*Binomial[k+n-1, n-1], {k, 1, n-m}]/(m-1)!, {m, 1, n}] + 1; Table[a[n], {n, 1, 17}] (* Jean-Fran├žois Alcover, Jul 05 2013, after Vladimir Kruchinin *)

PROG

(Maxima) a(n):=if n=1 then 1 else (n-1)!*sum(sum(binomial(k, n-m-k)* (-1/3)^(n-m-k)*binomial(k+n-1, n-1), k, 1, n-m)/(m-1)!, m, 1, n)+1; /* Vladimir Kruchinin, Aug 08 2010 */

CROSSREFS

Cf. A001515, A004747, A016036, A028575.

Sequence in context: A298691 A051442 A162650 * A290579 A140983 A241805

Adjacent sequences:  A015732 A015733 A015734 * A015736 A015737 A015738

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified July 22 09:46 EDT 2018. Contains 312891 sequences. (Running on oeis4.)