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A344396
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a(n) = binomial(2*n + 1, n)*hypergeom([-(n + 1)/2, -n/2], [n + 2], 4).
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1
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1, 5, 25, 133, 726, 4037, 22737, 129285, 740554, 4266830, 24701425, 143567173, 837212650, 4896136845, 28703894775, 168640510725, 992671051482, 5853000551090, 34562387229046, 204368928058958, 1209916827501876, 7170955214476509, 42543879586512435, 252638095187722437
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OFFSET
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0,2
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COMMENTS
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Related to the Motzkin triangle A064189 counting certain lattice paths.
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LINKS
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Table of n, a(n) for n=0..23.
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FORMULA
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a(n) = Sum_{j=0..2*n+1)} C(2*n + 1, j)*(C(2*n + 1 - j, j + n) - C(2*n + 1 - j, j + n + 2)).
a(n) = A064189(2*n+1, n).
a(n) = A026300(2*n+1, n+1).
a(n) ~ sqrt((5242 + 18674/sqrt(13))/2187) * ((70 + 26*sqrt(13))/27)^n / sqrt(Pi*n). - Vaclav Kotesovec, May 19 2021
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MAPLE
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alias(C=binomial):
a := n -> add(C(2*n + 1, j)*(C(2*n + 1 - j, j + n) - C(2*n + 1 - j, j + n + 2)), j = 0..2*n+1): seq(a(n), n=0..23);
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MATHEMATICA
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a[n_] := Binomial[2 n + 1, n] Hypergeometric2F1[-(n + 1)/2, -n/2, n + 2, 4];
Table[a[n], {n, 0, 23}]
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CROSSREFS
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Cf. A064189, A026300, A344394, A327871.
Sequence in context: A094602 A207834 A351187 * A351587 A225963 A222570
Adjacent sequences: A344393 A344394 A344395 * A344397 A344398 A344399
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KEYWORD
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nonn
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AUTHOR
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Peter Luschny, May 19 2021
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STATUS
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approved
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