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A350265
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a(n) = hypergeometric([-n - 1, 1 - n, -n], [1, 3], -1).
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1
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1, 1, 3, 12, 55, 276, 1477, 8296, 48393, 291010, 1794320, 11297760, 72413640, 471309944, 3108745785, 20746732688, 139899430981, 952127880138, 6533934575018, 45175430719240, 314467004704818, 2202576030828096, 15514620388706488, 109851319423632192, 781531332298053400
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) * A000217(n) = Sum_{k=0..n-1} binomial(n + 1, k) * binomial(n, k) * binomial(n + 1, k + 2).
a(n) * A002378(n) = Sum_{k=0..n-1} binomial(n + 1, k) * binomial(n + 1, k + 1) * binomial(n + 1, k + 2).
For a recurrence see the Maple program.
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MAPLE
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a := proc(n) option remember; if n < 2 then 1 else ((n + 1)*((7*n^2 + 7*n - 2)*a(n - 1) + 8*(n - 2)*n*a(n - 2)))/(n*(n + 2)*(n + 3)) fi end:
seq(a(n), n = 0..24);
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MATHEMATICA
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a[n_] := HypergeometricPFQ[{-n - 1, 1 - n, -n}, {1, 3}, -1];
Table[a[n], {n, 0, 24}]
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PROG
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(Python)
from sympy import hyperexpand
from sympy.functions import hyper
def A350265(n): return hyperexpand(hyper((-n-1, 1-n, -n), (1, 3), -1)) # Chai Wah Wu, Dec 29 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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