A015735 Row sums of triangle A004747.

1, 3, 17, 145, 1661, 23931, 415773, 8460257, 197360985, 5192853011, 152137882601, 4911873672113, 173268075672277, 6630323916472075, 273555262963272501, 12105084133976359361, 571897644855277242673, 28731255563712689630627, 1529450942687399074134465

Offset: 1

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Crossrefs

Cf. A001515, A004747, A016036, A028575.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-3*x)^(1/3)) - 1 ))); // G. C. Greubel, Oct 02 2023

Mathematica

a[1]=1; a[n_]:= 1 +(n-1)!*Sum[Binomial[k, n-m-k]*Binomial[k+n-1,n-1]*(-1/3)^(n-m-k)/(m-1)!, {m,n}, {k,n-m}]; Table[a[n], {n,20}] (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
Rest@With[{m=30}, CoefficientList[Series[Exp[1-Surd[1-3*x,3]] -1, {x, 0,m}], x]*Range[0,m]!] (* G. C. Greubel, Oct 02 2023 *)

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 */

SageMath

def A015735_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-3*x)^(1/3)) -1 ).egf_to_ogf().list()
a=A015735_list(40); a[1:] # G. C. Greubel, Oct 02 2023

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)*C(k+n-1, n-1)*(-1/3)^(n-m-k) ) / (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

From Seiichi Manyama, Jan 20 2025: (Start)

a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * |Stirling1(n,k)| * A000587(k).

a(n) = e * (-3)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/3,n)/k!. (End)