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%I #41 Oct 03 2023 13:16:48
%S 1,4,31,361,5626,109951,2585269,71066626,2236441141,79289379361,
%T 3127129674736,135802922499949,6439320471558781,331026965612789356,
%U 18338413238239145731,1089132347371148170381,69033182553940825258594
%N Row sums of triangle A000369.
%H Vincenzo Librandi, <a href="/A016036/b016036.txt">Table of n, a(n) for n = 1..200</a>
%H Wolfdieter Lang, <a href="https://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 - (1-4*x)^(1/4)) - 1.
%F a(n) = 6*(2*n-5)*a(n-1) - 3*(16*n^2-96*n+145)*a(n-2) + 2*(4*n-15)*(2*n-7)*(4*n-13)*a(n-3) + a(n-4), n >= 4; a(0) = 1, a(1) = 1, a(2) = 4, a(3) = 31.
%F a(n) = 1 + (n-1)!*Sum_{m=1..n-1} ( Sum_{k=1..n-m} binomial(n+k-1,n-1) * ( Sum_{j=0..k} binomial(j,n-m-3*k+2*j)*binomial(k,j)*3^(-n+m+3*k-j)*2^(n-m-5*k+3*j)*(-1)^(n-m-k) ) )/(m-1)!. - _Vladimir Kruchinin_, Oct 18 2011
%F a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^3*d/dx. Cf. A001515, A015735 and A028575. - _Peter Bala_, Nov 25 2011
%F a(n) ~ 2^(2*n-3/2)*n^(n-3/4)*exp(1-n)*sqrt(Pi)/Gamma(3/4) * (1 - Gamma(3/4)/(n^(1/4)*sqrt(Pi)) + Gamma(3/4)^2/(4*sqrt(n/2)*Pi)). - _Vaclav Kotesovec_, Aug 10 2013
%t a[n_, m_] /; (n>= m>= 1):= a[n, m]= (4*(n-1)-m)*a[n-1,m] + a[n-1,m-1]; a[n_, m_] /; n<m=0; a[_,0]= 0; a[1,1] = 1; a[n_]:= Sum[a[n,m], {m, n}]; Table[a[n], {n,20}] (* _Jean-François Alcover_, Feb 28 2013 *)
%t With[{nn=20},CoefficientList[Series[Exp[1-Surd[1-4x,4]]-1,{x,0,nn}],x] Range[0,nn]!]//Rest (* _Harvey P. Dale_, Apr 20 2016 *)
%o (Maxima)
%o a(n):=((n-1)!*sum((sum(binomial(n+k-1,n-1)*sum(binomial(j,n-m-3*k+2*j)*binomial(k,j)*3^(-n+m+3*k-j)*2^(n-m-5*k+3*j)*(-1)^(n-m-k),j,0,k),k,1,n-m))/(m-1)!,m,1,n-1))+1; /* _Vladimir Kruchinin_, Oct 18 2011 */
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-4*x)^(1/4)) -1 ))); // _G. C. Greubel_, Oct 02 2023
%o (SageMath)
%o def A016036_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(1-(1-4*x)^(1/4)) -1 ).egf_to_ogf().list()
%o a=A016036_list(40); a[1:] # _G. C. Greubel_, Oct 02 2023
%Y Sequences with e.g.f. exp(1-(1-m*x)^(1/m)) - 1: A000012 (m=1), A001515 (m=2), A015735 (m=3), this sequence (m=4), A028575 (m=5), A028844 (m=6).
%Y Cf. A000369.
%K nonn
%O 1,2
%A _Wolfdieter Lang_
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