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
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Mark Shattuck, Enumeration of consecutive patterns in flattened Catalan words, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 6.
- Index entries for linear recurrences with constant coefficients, signature (8,-22,24,-9).
Programs
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GAP
List([0..10^2], n->Sum([1..n], k->Sum([1..3], j->Stirling2(n,j)))); # Muniru A Asiru, Feb 06 2018
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Magma
[n*(1^(n-1) + 3^(n-1))/2: n in [0..30]]; // G. C. Greubel, Feb 05 2018
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Maple
with (combinat):seq(sum(sum(stirling2(n, j),j=1..3), k=1..n), n=0..24); # Zerinvary Lajos, Dec 04 2007
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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 *) -
PARI
for(n=0,30, print1(n*(1^(n-1) + 3^(n-1))/2, ", ")) \\ G. C. Greubel, Feb 05 2018
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
Extensions
Definition clarified by Harvey P. Dale, Apr 30 2012
Comments