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A302352 a(n) = Sum_{k=0..n} k^4*binomial(2*n-k,n). 3
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A355217 A254142 A107891 * A301398 A141923 A261791
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 06 2018
STATUS
approved

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Last modified June 19 00:22 EDT 2024. Contains 373492 sequences. (Running on oeis4.)