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A163774
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Row sums of the central coefficients triangle (A163771).
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3
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1, 3, 13, 51, 201, 783, 3039, 11763, 45481, 175803, 679779, 2630367, 10187659, 39500373, 153329913, 595883763, 2318471289, 9030982491, 35216266947, 137469149451, 537152523711, 2100857828193, 8223917499477, 32219655346719, 126328429601451, 495676719721953, 1946227355491909
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$, where i$ denotes the swinging factorial of i (A056040).
Conjecture: a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+1,k)*binomial(2*k,k). - Werner Schulte, Nov 17 2015
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MAPLE
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swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i, k; add(add((-1)^(n-i)*binomial(n-k, n-i)*swing(2*i), i=k..n), k=0..n) end:
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MATHEMATICA
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sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)
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CROSSREFS
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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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