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A339710
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a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n + k, k)*2^k.
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6
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1, 7, 81, 1051, 14353, 201807, 2891409, 41976627, 615371169, 9089130967, 135048608401, 2016306678987, 30224723308081, 454603719479839, 6857319231939537, 103694587800440931, 1571449259865571137, 23860205774602899111, 362897293035114695121, 5527773456878667951483
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OFFSET
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0,2
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REFERENCES
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Frits Beukers, Some Congruences for Apery Numbers, Mathematisch Instituut, University of Leiden, 1983, pages 1-2.
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LINKS
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FORMULA
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a(n) = 2F1([-n, 1 + 2*n], [1], -2), where 2F1 is the hypergeometric function. - Stefano Spezia, Dec 17 2020
Recurrence: 3*n*(2*n - 1)*(26*n - 35)*a(n) = (2444*n^3 - 5734*n^2 + 3830*n - 729)*a(n-1) - (n-1)*(2*n - 3)*(26*n - 9)*a(n-2).
a(n) ~ sqrt(3/8 + 11/(8*sqrt(13))) * ((47 + 13*sqrt(13))/6)^n / sqrt(Pi*n). (End)
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MATHEMATICA
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Table[Sum[Binomial[n, k]*Binomial[2n+k, k]*2^k, {k, 0, n}], {n, 0, 20}] (* or *)
Table[Hypergeometric2F1[-n, 1+2 n, 1, -2], {n, 0, 20}] (* Stefano Spezia, Dec 17 2020 *)
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n + k, k)*2^k); \\ Michel Marcus, Feb 12 2021
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CROSSREFS
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Cf. A000079 (Sum(binomial(n, k))), A000984 (Sum(binomial(n, k)^2)), A026375 (Sum(binomial(n, k)*binomial(2*k, k))), A001850 (Sum(binomial(n, k)*binomial(n+k, k))), A005809 (Sum(binomial(n, k)*binomial(2*n, k))).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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