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A264960 Half-convolution of the central binomial coefficients A000984 with itself. 3
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 (list; graph; refs; listen; history; text; internal format)
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 ).

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

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Last modified March 7 10:14 EST 2021. Contains 341869 sequences. (Running on oeis4.)