A016036 Row sums of triangle A000369.

1, 4, 31, 361, 5626, 109951, 2585269, 71066626, 2236441141, 79289379361, 3127129674736, 135802922499949, 6439320471558781, 331026965612789356, 18338413238239145731, 1089132347371148170381, 69033182553940825258594

Offset

1,2

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.

Formula

E.g.f.: exp(1 - (1-4*x)^(1/4)) - 1.

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.

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

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

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

Mathematica

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 *)
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 *)

Prog

(Maxima)
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 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-4*x)^(1/4)) -1 ))); // G. C. Greubel, Oct 02 2023
(SageMath)
def A016036_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-4*x)^(1/4)) -1 ).egf_to_ogf().list()
a=A016036_list(40); a[1:] # G. C. Greubel, Oct 02 2023

Crossrefs

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).

Cf. A000369.

Sequence in context: A201628 A086677 A368237 * A322626 A000314 A128709

Adjacent sequences: A016033 A016034 A016035 * A016037 A016038 A016039

Keyword

nonn

Author

Wolfdieter Lang

Status

approved