login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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).
(Formerly M4656)
5
1, 9, 92, 920, 8928, 84448, 782464, 7130880, 64117760, 570166784, 5023524864, 43915595776, 381350330368, 3292451880960, 28283033157632, 241884640182272, 2060565937127424, 17492250190544896, 148027589475696640 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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

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 *)

PROG

(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

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

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.

Sequence in context: A225162 A128384 A164913 * A015587 A024117 A261855

Adjacent sequences:  A007400 A007401 A007402 * A007404 A007405 A007406

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Mira Bernstein

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 1 04:57 EDT 2020. Contains 333155 sequences. (Running on oeis4.)