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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)
10
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
(PARI) a(n)=(n+2)<<(3*n-1)-3*n*binomial(2*n, n)<<(n-2) \\ Charles R Greathouse IV, Oct 23 2023
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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)