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A264960 Half-convolution of the central binomial coefficients A000984 with itself.

%I A264960 #23 Mar 05 2025 05:25:27
%S A264960 1,2,10,32,146,512,2248,8192,35218,131072,556040,2097152,8815496,
%T A264960 33554432,140107040,536870912,2230302098,8589934592,35541690568,
%U A264960 137438953472,566823203656,2199023255552,9044910175520,35184372088832,144393718191496
%N A264960 Half-convolution of the central binomial coefficients A000984 with itself.
%C A264960 The half-convolution of a sequence {s(n)}n>=0 with itself is defined by r(n) := Sum_{k = 0..floor(n/2)} s(k)*s(n-k). See A201204.
%H A264960 Muniru A Asiru, <a href="/A264960/b264960.txt">Table of n, a(n) for n = 0..1520</a>
%F A264960 a(n) = Sum_{k = 0..floor(n/2)} binomial(2*k,k)*binomial(2*n - 2*k, n - k).
%F A264960 a(2*n + 1) = 2^(4*n + 1) = A013776(n).
%F A264960 a(2*n) = (1/2)*(binomial(2*n,n)^2 + 16^n) = A112830(2*n,n).
%F A264960 O.g.f.: (1/2)*( 2/Pi*EllipticK(4*x) + 1/(1 - 4*x) ).
%F A264960 E.g.f.: (1/2)*( cosh(4*x) + sinh(4*x) + (BesselI(0,2*x))^2 ).
%F A264960 D-finite with recurrence: - (2*n-3)*n^2*a(n) + 4*(2*n-1)*(n-1)^2*a(n-1) + 16*(2*n-3)*(n-1)^2*a(n-2) - 64*(2*n-1)*(n-2)^2*a(n-3) = 0. - _Georg Fischer_, Nov 25 2022
%p A264960 A264960:= n-> add(binomial(2*k,k)*binomial(2*n - 2*k, n - k),k = 0..floor(n/2)):
%p A264960 seq(A264960(n),n = 0..24);
%t A264960 a[n_] := Sum[Binomial[2k, k]*Binomial[2n - 2k, n - k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* _Amiram Eldar_, Nov 25 2018 *)
%o A264960 (PARI) a(n) = sum(k = 0, n\2, binomial(2*k,k)*binomial(2*n - 2*k, n - k)); \\ _Michel Marcus_, Nov 30 2015
%o A264960 (GAP) List([0..24],n->Sum([0..Int(n/2)],k->Binomial(2*k,k)*Binomial(2*n-2*k,n-k))); # _Muniru A Asiru_, Nov 25 2018
%o A264960 (Magma) [(&+[Binomial(2*k,k)*Binomial(2*n-2*k, n-k): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Nov 26 2018
%o A264960 (Sage) [sum(binomial(2*k,k)*binomial(2*n-2*k, n-k) for k in (0..floor(n/2))) for n in range(30)] # _G. C. Greubel_, Nov 26 2018
%Y A264960 Cf. A002894, A013776, A112830.
%K A264960 nonn,easy
%O A264960 0,2
%A A264960 _Peter Bala_, Nov 29 2015