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

1, 2, 10, 32, 146, 512, 2248, 8192, 35218, 131072, 556040, 2097152, 8815496, 33554432, 140107040, 536870912, 2230302098, 8589934592, 35541690568, 137438953472, 566823203656, 2199023255552, 9044910175520, 35184372088832, 144393718191496

Offset

0,2

Comments

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.

Links

Muniru A Asiru, Table of n, a(n) for n = 0..1520

Formula

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

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

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

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

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

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

Maple

A264960:= n-> add(binomial(2*k, k)*binomial(2*n - 2*k, n - k), k = 0..floor(n/2)):
seq(A264960(n), n = 0..24);

Mathematica

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 *)

Prog

(PARI) a(n) = sum(k = 0, n\2, binomial(2*k, k)*binomial(2*n - 2*k, n - k)); \\ Michel Marcus, Nov 30 2015
(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
(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
(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

Crossrefs

Cf. A002894, A013776, A112830.

Sequence in context: A083099 A032095 A328039 * A151019 A329427 A004028

Adjacent sequences: A264957 A264958 A264959 * A264961 A264962 A264963

Keyword

nonn,easy

Author

Peter Bala, Nov 29 2015

Status

approved