A302352 a(n) = Sum_{k=0..n} k^4*binomial(2*n-k,n).

0, 1, 19, 155, 936, 4884, 23465, 107107, 472600, 2036838, 8631206, 36119798, 149724940, 616104450, 2520629685, 10265200035, 41650094640, 168481778790, 679847488650, 2737640364810, 11005139655744, 44176226269728, 177114113623194, 709364594864910, 2838599638596176, 11350436081373340

Offset

0,3

Comments

Main diagonal of iterated partial sums array of fourth powers (starting with the first partial sums). For nonnegative integers see A002054, for squares see A265612, for cubes see A293550.

Links

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

Eric Weisstein's World of Mathematics, Biquadratic Number

Formula

a(n) = [x^n] x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^(n+6).

a(n) = 2^(2*n+1)*n*(75*n^3 + 52*n^2 - 3*n - 4)*Gamma(n+3/2)/(sqrt(Pi)*Gamma(n+6)).

a(n) ~ 75*2^(2*n+1)/sqrt(Pi*n).

Mathematica

Table[Sum[k^4 Binomial[2 n - k, n], {k, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[x (1 + 11 x + 11 x^2 + x^3)/(1 - x)^(n + 6), {x, 0, n}], {n, 0, 25}]
Table[2^(2 n + 1) n (75 n^3 + 52 n^2 - 3 n - 4) Gamma[n + 3/2]/(Sqrt[Pi] Gamma[n + 6]), {n, 0, 25}]
CoefficientList[Series[(24 - 180 x + 410 x^2 - 285 x^3 + 31 x^4 + Sqrt[1 - 4 x] (-24 + 132 x - 194 x^2 + 65 x^3 - x^4))/(2 Sqrt[1 - 4 x] x^5), {x, 0, 25}], x]
CoefficientList[Series[E^(2 x) (-576 + 360 x - 244 x^2 + 75 x^3) BesselI[0, 2 x]/x^3 + E^(2 x) (576 - 360 x + 532 x^2 - 255 x^3 + 75 x^4) BesselI[1, 2 x]/x^4, {x, 0, 25}], x]* Range[0, 25]!

Prog

(PARI) a(n) = sum(k=0, n, k^4*binomial(2*n-k, n)); \\ Michel Marcus, Apr 07 2018

Crossrefs

Cf. A000538, A000583, A002054, A101089, A101090, A101091, A265612, A293550, A302353.

Sequence in context: A355217 A254142 A107891 * A301398 A141923 A261791

Adjacent sequences: A302349 A302350 A302351 * A302353 A302354 A302355

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 06 2018

Status

approved