A089664 a(n) = S2(n,1), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

0, 4, 41, 306, 1966, 11540, 63726, 336700, 1720364, 8562024, 41718190, 199753004, 942561636, 4392660376, 20253510956, 92519626200, 419201709976, 1885719209936, 8428262686254, 37453751742604, 165575219275700, 728534225415864, 3191850894862564

Offset: 0

Author

N. J. A. Sloane, Jan 04 2004

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Crossrefs

Sequences of S2(n, t): A003583 (t=0), this sequence (t=1), A089665 (t=2), A089666 (t=3), A089667 (t=4), A089668 (t=5).

Cf. A000108, A089658, A089669.

Programs

Mathematica

Table[(n*(3*n+5)*4^n -2*n*(n-1)*Binomial[2*n,n])/8, {n,0,40}] (* G. C. Greubel, May 25 2022 *)

PARI

a(n)=n*(3*n+5)*2^(2*n-3) - 3*binomial(n+1,3)*binomial(2*n,n)/(n+1)/2 \\ Charles R Greathouse IV, Oct 23 2023

SageMath

[(1/2)*(n*(3*n+5)*4^(n-1) -3*binomial(n+1, 3)*catalan_number(n)) for n in (0..40)] # G. C. Greubel, May 25 2022

Formula

a(n) = (1/8)*(n*(3*n+5)*4^n - 2*n*(n-1)*binomial(2*n, n)). (see Wang and Zhang, p. 338)

From G. C. Greubel, May 25 2022: (Start)

a(n) = (1/2)*( n*(3*n+5)*4^(n-1) - 3*binomial(n+1, 3)*Catalan(n) ).

G.f.: x*(4*(1-x) - 3*x*sqrt(1-4*x))/(1-4*x)^3.

E.g.f.: 2*x*(2 + 3*x)*exp(4*x) - (x^2/2)*(3*BesselI(0, 2*x) + 4*BesselI(1, 2*x) + BesselI(2, 2*x))*exp(2*x)). (End)