A293550 a(n) = Sum_{k=0..n} k^3*binomial(2*n-k,n).

0, 1, 11, 69, 354, 1650, 7293, 31213, 130832, 540702, 2212550, 8989090, 36327810, 146228940, 586823265, 2349424125, 9389012160, 37467344310, 149345215290, 594753416790, 2366845396500, 9413555798556, 37423053793026, 148719333293394, 590842248405024, 2346813893147500

Offset

0,3

Comments

Main diagonal of iterated partial sums array of cubes (starting with the first partial sums). For nonnegative integers see A002054, for squares see A265612.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for sequences related to sums of cubes

Formula

a(n) = [x^n] x*(1 + 4*x + x^2)/(1 - x)^(n+5).

a(n) = 2^(2*n+1)*n^2*(13*n + 7)*Gamma(n+3/2)/(sqrt(Pi)*Gamma(n+5)).

a(n) ~ 26*4^n/sqrt(Pi*n).

Mathematica

Table[Sum[k^3 Binomial[2 n - k, n], {k, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[x (1 + 4 x + x^2)/(1 - x)^(n + 5), {x, 0, n}], {n, 0, 25}]
Table[2^(2 n + 1) n^2 (13 n + 7) Gamma[n + 3/2]/(Sqrt[Pi] Gamma[n + 5]), {n, 0, 25}]
CoefficientList[Series[(6 - 6 Sqrt[1 - 4 x] - 36 x + 24 Sqrt[1 - 4 x] x + 55 x^2 - 19 Sqrt[1 - 4 x] x^2 - 15 x^3 + Sqrt[1 - 4 x] x^3)/(2 Sqrt[1 - 4 x] x^4), {x, 0, 25}], x]
CoefficientList[Series[(E^(2 x) (36 - 24 x + 13 x^2) BesselI[0, 2 x])/x^2 + (E^(2 x) (-36 + 24 x - 31 x^2 + 13 x^3) BesselI[1, 2 x])/x^3, {x, 0, 25}], x]* Range[0, 25]!

Crossrefs

Cf. A000537, A000578, A002054, A024166, A101094, A101097, A101102, A265612, A302352.

Sequence in context: A292490 A169731 A212057 * A182188 A099336 A236320

Adjacent sequences: A293547 A293548 A293549 * A293551 A293552 A293553

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 11 2017

Status

approved