A007403 a(n) = Sum_{m=0..n} (Sum_{k=0..m} binomial(n,k))^3 = (n+2)*2^(3*n-1) - 3*2^(n-2)*n*binomial(2*n,n).

1, 9, 92, 920, 8928, 84448, 782464, 7130880, 64117760, 570166784, 5023524864, 43915595776, 381350330368, 3292451880960, 28283033157632, 241884640182272, 2060565937127424, 17492250190544896, 148027589475696640

Offset: 0

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• nice

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1104 (terms 0..200 from Vincenzo Librandi)
• G. E. Andrews and P. Paule, MacMahon's partition analysis. IV. Hypergeometric multisums, In The Andrews Festschrift (Maratea, 1998). Sem. Lothar. Combin. 42 (1999), Art. B42i, 24 pp.
• N. J. Calkin, A curious binomial identity, Discr. Math., 131 (194), 335-337.
• Bing He, Some identities involving the partial sum of q-binomial coefficients, Electronic J. Combin,, 21 (2014), #P3.17. Gives generalizations. - _N. J. A. Sloane_, Jul 26 2014
• M. Hirschhorn, Calkin's binomial identity, Discr. Math., 159 (1996), 273-278.
• C. L. Mallows, Letter to N. J. Calkin [Included with permission]
• Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Crossrefs

If the exponent E in a(n) = Sum_{m=0..n} (Sum_{k=0..m} C(n,k))^E is 1, 2, 3, 4, 5 we get A001792, A003583, A007403, A294435, A294436 respectively.

Programs

GAP

List([0..20],n->Sum([0..n],m->Sum([0..m],k->Binomial(n,k))^3)); # Muniru A Asiru, Aug 15 2018

Magma

[(n+2)*2^(3*n-1)-3*2^(n-2)*n*Binomial(2*n,n): n in [0..20]]; // Vincenzo Librandi, Jul 27 2014

Maple

f:=n->n*8^n/2+8^n-(3*n/4)*2^n*binomial(2*n,n);
[seq(f(n),n=0..50)];
A:=proc(n,k) local j; add(binomial(n,j),j=0..k); end;
S:=proc(n,p) local i; global A; add(A(n,i)^p, i=0..n); end;
[seq(S(n,3),n=0..50)]; # N. J. A. Sloane, Nov 17 2017

Mathematica

Table[(n+2)2^(3n-1)-3 2^(n-2)n Binomial[2n,n],{n,0,20}] (* Harvey P. Dale, Jun 30 2011 *)
CoefficientList[Series[(1 - (4 + 3 Sqrt[1 - 8 x]) x)/(1 - 8 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Jul 27 2014 *)
nmax = 18; CoefficientList[Series[Exp[8 x] (1 + 4 x) - 3 x Exp[4 x] (BesselI[0, 4 x] + BesselI[1, 4 x]), {x, 0, nmax}], x] Range[0, nmax]! (* Ilya Gutkovskiy, Aug 18 2018 *)

PARI

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

Formula

G.f.: (1 - (4 + 3*sqrt(1 - 8*x))*x)/(1 - 8*x)^2. - Harvey P. Dale, Jun 30 2011

E.g.f.: exp(8*x)*(1 + 4*x) - 3*x*exp(4*x)*(BesselI(0,4*x) + BesselI(1,4*x)). - Ilya Gutkovskiy, Aug 15 2018

a(n) ~ n * 2^(3*n-1) * (1 - 3/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Aug 18 2018