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

Author

N. J. A. Sloane, Jun 13 2013

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

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

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 *)

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