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 A293550 a(n) = Sum_{k=0..n} k^3*binomial(2*n-k,n). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified October 18 22:03 EDT 2019. Contains 328211 sequences. (Running on oeis4.)