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A249608
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Expansion of e.g.f.: exp(x) * BesselI(0, 2*x) * BesselI(0, 2*sqrt(2)*x).
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1
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1, 1, 7, 19, 115, 451, 2521, 11677, 63379, 318115, 1716517, 8981017, 48623389, 260410333, 1418640055, 7707719299, 42263782099, 231857347603, 1278917211061, 7065478018585, 39174182961865, 217591443710905, 1211736702238795, 6759723716824855, 37785032547293245, 211540175726995501
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^k * C(n,k) * C(2*k,k) * (Sum_{j=0..n-k} C(n-k,j)^2 * 2^j).
a(n) = Sum_{k=0..n} (-1)^k * C(n,k) * C(2*k,k) * A001850(n-k), where A001850 is the central Delannoy numbers.
D-finite with recurrence: (n-2)*n^2*a(n) = (4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) + (n-1)*(18*n^2 - 54*n + 35)*a(n-2) - 44*(n-2)^2*(n-1)*a(n-3) + 7*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Nov 09 2014
a(n) ~ (3+2*sqrt(2))^(n+1) / (2^(9/4) * Pi * n). - Vaclav Kotesovec, Nov 09 2014
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EXAMPLE
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E.g.f.: E(x) = 1 + x + 7*x^2/2! + 19*x^3/3! + 115*x^4/4! + 451*x^5/5! + ...
such that E(x) = exp(x)*F(x)*G(x) where
F(x) = 1 + 2*x^2/2! + 6*x^4/4! + 20*x^6/6! + 70*x^8/8! + 252*x^10/10! + ...
G(x) = 1 + 4*x^2/2! + 24*x^4/4! + 160*x^6/6! + 1120*x^8/8! + 8064*x^10/10! + ...
O.g.f.: A(x) = 1 + x + 7*x^2 + 19*x^3 + 115*x^4 + 451*x^5 + 2521*x^6 + ...
where
x/Series_Reversion(x*A(x)) = 1 + x + 6*x^2 + 6*x^4 - 36*x^6 - 138*x^8 + 1044*x^10 + 2364*x^12 - 25416*x^14 - 89946*x^16 + 921348*x^18 + 2457972*x^20 + ...
(x/Series_Reversion(x*A(x)) - x)^2 = 1 + 12*x^2 + 48*x^4 - 672*x^8 + 16896*x^12 - 512640*x^16 + 17160192*x^20 - 610750464*x^24 + 22672244736*x^28 + ...
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MAPLE
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seq(coeff(series(factorial(n)*exp(x)*BesselI(0, 2*x)*BesselI(0, 2*x*sqrt(2)), x, n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 08 2018
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MATHEMATICA
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Table[Sum[(-1)^k * Binomial[n, k] * Binomial[2*k, k] * Sum[Binomial[n-k, j]^2 * 2^j, {j, 0, n-k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 09 2014 *)
CoefficientList[Series[E^x * BesselI[0, 2*x] * BesselI[0, 2*Sqrt[2]*x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Nov 09 2014 *)
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PROG
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(PARI) /* Using product of Bessel functions: */
{a(n)=local(A=1, B1=sum(m=0, n, x^(2*m)/m!^2)+x*O(x^n), B2=sum(m=0, n, 2^m*x^(2*m)/m!^2)+x*O(x^n));
n!*polcoeff(exp(x +x*O(x^n))*B1*B2, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Using binomial sum: */
{a(n)=sum(k=0, n, (-1)^k*binomial(n, k)*binomial(2*k, k)*sum(j=0, n-k, binomial(n-k, j)^2*2^j))}
for(n=0, 30, print1(a(n), ", "))
(Magma) [(&+[(&+[(-1)^k*Binomial(n, k)*Binomial(2*k, k)*Binomial(n-k, j)^2*2^j: j in [0..n-k]]):k in [0..n]]): n in [0..20]]; // G. C. Greubel, Oct 07 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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