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A163774 Row sums of the central coefficients triangle (A163771).

%I A163774 #22 Aug 22 2025 04:25:34
%S A163774 1,3,13,51,201,783,3039,11763,45481,175803,679779,2630367,10187659,
%T A163774 39500373,153329913,595883763,2318471289,9030982491,35216266947,
%U A163774 137469149451,537152523711,2100857828193,8223917499477,32219655346719,126328429601451,495676719721953,1946227355491909
%N A163774 Row sums of the central coefficients triangle (A163771).
%H A163774 G. C. Greubel, <a href="/A163774/b163774.txt">Table of n, a(n) for n = 0..1000</a>
%H A163774 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163774 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163774 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).
%F A163774 Conjecture: a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+1,k)*binomial(2*k,k). - _Werner Schulte_, Nov 17 2015
%p A163774 swing := proc(n) option remember; if n = 0 then 1 elif
%p A163774 irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p A163774 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:
%t A163774 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 *)
%Y A163774 Cf. A056040, A163771.
%K A163774 nonn,changed
%O A163774 0,2
%A A163774 _Peter Luschny_, Aug 05 2009
%E A163774 More terms from _Michel Marcus_, Nov 24 2015