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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 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

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