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A264960 Half-convolution of the central binomial coefficients A000984 with itself. 3

%I

%S 1,2,10,32,146,512,2248,8192,35218,131072,556040,2097152,8815496,

%T 33554432,140107040,536870912,2230302098,8589934592,35541690568,

%U 137438953472,566823203656,2199023255552,9044910175520,35184372088832,144393718191496

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

%C 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 Muniru A Asiru, <a href="/A264960/b264960.txt">Table of n, a(n) for n = 0..1520</a>

%F a(n) = Sum_{k = 0..floor(n/2)} binomial(2*k,k)*binomial(2*n - 2*k, n - k).

%F a(2*n + 1) = 2^(4*n + 1) = A013776(n).

%F a(2*n) = 1/2*(binomial(2*n,n)^2 + 16^n) = A112830(2*n,n).

%F O.g.f.: 1/2*( 2/Pi*EllipticK(4*x)) + 1/(1 - 4*x) ).

%F E.g.f.: 1/2*( cosh(4*x) + sinh(4*x) + (BesselI(0,2*x))^2 ).

%p A264960:= n-> add(binomial(2*k,k)*binomial(2*n - 2*k, n - k),k = 0..floor(n/2)):

%p seq(A264960(n),n = 0..24);

%t 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 (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 (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 (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 (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 Cf. A002894, A013776, A112830.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Nov 29 2015

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Last modified April 13 06:02 EDT 2021. Contains 342935 sequences. (Running on oeis4.)