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%I #47 Oct 02 2023 13:17:10
%S 1,3,17,145,1661,23931,415773,8460257,197360985,5192853011,
%T 152137882601,4911873672113,173268075672277,6630323916472075,
%U 273555262963272501,12105084133976359361,571897644855277242673
%N Row sums of triangle A004747.
%H Vincenzo Librandi, <a href="/A015735/b015735.txt">Table of n, a(n) for n = 1..200</a>
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F E.g.f.: exp(1-(1-3*x)^(1/3)) - 1, if one takes a(0)=0.
%F 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.
%F 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
%F 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
%F E.g.f. with offset 0: exp(1-(1-3*x)^(1/3))/(1-3*x)^(2/3). - _Sergei N. Gladkovskii_, Jul 07 2012.
%F 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
%t 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_ *)
%t 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 *)
%o (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 */
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-3*x)^(1/3)) - 1 ))); // _G. C. Greubel_, Oct 02 2023
%o (SageMath)
%o def A015735_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(1-(1-3*x)^(1/3)) -1 ).egf_to_ogf().list()
%o a=A015735_list(40); a[1:] # _G. C. Greubel_, Oct 02 2023
%Y Cf. A001515, A004747, A016036, A028575.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_
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