A082133 Expansion of e.g.f. x*exp(2*x)*cosh(x).

0, 1, 4, 15, 56, 205, 732, 2555, 8752, 29529, 98420, 324775, 1062888, 3454373, 11160268, 35872275, 114791264, 365897137, 1162261476, 3680494655, 11622614680, 36611236221, 115063885244, 360882185515, 1129718145936

Offset

0,3

Comments

Binomial transform of A057711. 2nd binomial transform of (0,1,0,3,0,5,0,7,...).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-22,24,-9).

Formula

a(n) = n*(1^(n-1) + 3^(n-1))/2.

E.g.f.: x*exp(2x)*cosh(x).

G.f.: x*(1-4*x+5*x^2) / ( (3*x-1)^2*(x-1)^2 ). - R. J. Mathar, Nov 24 2012

a(n) = Sum_{k=1..n} (Sum_{j=1..3} Stirling2(n,j)). - G. C. Greubel, Feb 07 2018

Maple

with (combinat):seq(sum(sum(stirling2(n, j), j=1..3), k=1..n), n=0..24); # Zerinvary Lajos, Dec 04 2007

Mathematica

With[{nn=30}, CoefficientList[Series[x Exp[2x]Cosh[x], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Apr 30 2012 *)
Table[n*(1^(n-1) + 3^(n-1))/2, {n, 0, 30}] (* G. C. Greubel, Feb 05 2018 *)
Table[Sum[Sum[StirlingS2[n, j], {j, 1, 3}], {k, 1, n}], {n, 0, 30}] (* G. C. Greubel, Feb 07 2018 *)

Prog

(PARI) for(n=0, 30, print1(n*(1^(n-1) + 3^(n-1))/2, ", ")) \\ G. C. Greubel, Feb 05 2018
(Magma) [n*(1^(n-1) + 3^(n-1))/2: n in [0..30]]; // G. C. Greubel, Feb 05 2018
(GAP) List([0..10^2], n->Sum([1..n], k->Sum([1..3], j->Stirling2(n, j)))); # Muniru A Asiru, Feb 06 2018

Crossrefs

Cf. A082134, A082135, A082136.

Sequence in context: A183932 A009940 A081163 * A060111 A077824 A291030

Adjacent sequences: A082130 A082131 A082132 * A082134 A082135 A082136

Keyword

easy,nonn

Author

Paul Barry, Apr 06 2003

Extensions

Definition clarified by Harvey P. Dale, Apr 30 2012

Status

approved