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%I M4656 #58 Oct 23 2023 11:13:39
%S 1,9,92,920,8928,84448,782464,7130880,64117760,570166784,5023524864,
%T 43915595776,381350330368,3292451880960,28283033157632,
%U 241884640182272,2060565937127424,17492250190544896,148027589475696640
%N 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).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A007403/b007403.txt">Table of n, a(n) for n = 0..1104</a> (terms 0..200 from Vincenzo Librandi)
%H G. E. Andrews and P. Paule, <a href="https://www.emis.de/journals/SLC/wpapers/s42paule.html">MacMahon's partition analysis. IV. Hypergeometric multisums</a>, In The Andrews Festschrift (Maratea, 1998). Sem. Lothar. Combin. 42 (1999), Art. B42i, 24 pp.
%H N. J. Calkin, <a href="http://dx.doi.org/10.1016/0012-365X(94)90394-8">A curious binomial identity</a>, Discr. Math., 131 (194), 335-337.
%H Bing He, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p17">Some identities involving the partial sum of q-binomial coefficients</a>, Electronic J. Combin,, 21 (2014), #P3.17. Gives generalizations. - _N. J. A. Sloane_, Jul 26 2014
%H M. Hirschhorn, <a href="http://dx.doi.org/10.1016/0012-365X(95)00086-C">Calkin's binomial identity</a>, Discr. Math., 159 (1996), 273-278.
%H C. L. Mallows, <a href="/A007403/a007403.png">Letter to N. J. Calkin</a> [Included with permission]
%H Jun Wang and Zhizheng Zhang, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00206-1">On extensions of Calkin's binomial identities</a>, Discrete Math., 274 (2004), 331-342.
%F G.f.: (1 - (4 + 3*sqrt(1 - 8*x))*x)/(1 - 8*x)^2. - _Harvey P. Dale_, Jun 30 2011
%F 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
%F a(n) ~ n * 2^(3*n-1) * (1 - 3/(2*sqrt(Pi*n))). - _Vaclav Kotesovec_, Aug 18 2018
%p f:=n->n*8^n/2+8^n-(3*n/4)*2^n*binomial(2*n,n);
%p [seq(f(n),n=0..50)];
%p A:=proc(n,k) local j; add(binomial(n,j),j=0..k); end;
%p S:=proc(n,p) local i; global A; add(A(n,i)^p, i=0..n); end;
%p [seq(S(n,3),n=0..50)]; # _N. J. A. Sloane_, Nov 17 2017
%t Table[(n+2)2^(3n-1)-3 2^(n-2)n Binomial[2n,n],{n,0,20}] (* _Harvey P. Dale_, Jun 30 2011 *)
%t CoefficientList[Series[(1 - (4 + 3 Sqrt[1 - 8 x]) x)/(1 - 8 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Jul 27 2014 *)
%t 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 *)
%o (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
%o (GAP) List([0..20],n->Sum([0..n],m->Sum([0..m],k->Binomial(n,k))^3)); # _Muniru A Asiru_, Aug 15 2018
%o (PARI) a(n)=(n+2)<<(3*n-1)-3*n*binomial(2*n,n)<<(n-2) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y 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.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, _Mira Bernstein_
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