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A016036 Row sums of triangle A000369. 7
1, 4, 31, 361, 5626, 109951, 2585269, 71066626, 2236441141, 79289379361, 3127129674736, 135802922499949, 6439320471558781, 331026965612789356, 18338413238239145731, 1089132347371148170381, 69033182553940825258594 (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-(1-4*x)^(1/4))-1.

Recursion: 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) = ((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)!))+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

a(n) = 6*(2*n-5)*a(n-1) - 3*(16*n^2-96*n+145)*a(n-2) + 2*(2*n-7)*(4*n-15)*(4*n-13)*a(n-3) + a(n-4). - 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, 1, n}]; Table[a[n], {n, 1, 17}] (* Jean-Fran├žois 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 */

CROSSREFS

Cf. A001515, A015735.

Sequence in context: A145561 A201628 A086677 * A322626 A000314 A128709

Adjacent sequences:  A016033 A016034 A016035 * A016037 A016038 A016039

KEYWORD

nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified April 19 16:18 EDT 2019. Contains 322282 sequences. (Running on oeis4.)