A226514 Column 3 of array in A226513.

13, 44, 99, 184, 305, 468, 679, 944, 1269, 1660, 2123, 2664, 3289, 4004, 4815, 5728, 6749, 7884, 9139, 10520, 12033, 13684, 15479, 17424, 19525, 21788, 24219, 26824, 29609, 32580, 35743, 39104, 42669, 46444, 50435, 54648, 59089, 63764, 68679, 73840, 79253, 84924, 90859, 97064, 103545, 110308, 117359, 124704

Offset

0,1

Comments

This is the case h=3 in sum( S2(h,k)*k!*binomial(n+k,k), k=0..h ), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [Bruno Berselli, Jun 20 2013]

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = (n + 1)*(n^2 + 8*n + 13).

G.f.: (13 - 8*x + x^2)/(1 - x)^4. - Vincenzo Librandi, Jun 17 2013

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 17 2013

E.g.f.: exp(x)*(13 + 31*x + 12*x^2 + x^3). - Franck Maminirina Ramaharo, Nov 29 2018

Mathematica

Table[n^3 + 9*n^2 + 21*n + 13, {n, 0, 50}] (* or *) CoefficientList[Series[(13 - 8 x + x^2) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 17 2013 *)

Prog

(Magma) [n^3+9*n^2+21*n+13: n in [0..50]]; // Vincenzo Librandi, Jun 17 2013
(Magma) I:=[13, 44, 99, 184]; [n le 4 select I[n] else 4*Self(n-1) -6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 17 2013

Crossrefs

Cf. columns 2 and 4, 5, 6 of A226513: A005563, A226741, A226800, A226801.

Sequence in context: A015247 A033652 A026914 * A159742 A098385 A048364

Adjacent sequences: A226511 A226512 A226513 * A226515 A226516 A226517

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 13 2013

Status

approved