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A007403
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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)
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10
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1, 9, 92, 920, 8928, 84448, 782464, 7130880, 64117760, 570166784, 5023524864, 43915595776, 381350330368, 3292451880960, 28283033157632, 241884640182272, 2060565937127424, 17492250190544896, 148027589475696640
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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