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A098409
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Expansion of 1/(sqrt(1-3*x)*sqrt(1-7*x)).
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14
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1, 5, 27, 155, 931, 5775, 36645, 236325, 1542195, 10153775, 67313377, 448691985, 3004182349, 20188647185, 136094684907, 919884469275, 6232016686995, 42305974804575, 287706424085745, 1959685788407025, 13367193276457881, 91295551930615005, 624255065007468207
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OFFSET
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0,2
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COMMENTS
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Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in five colors. - N-E. Fahssi, Feb 05 2008
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have five colors. - N-E. Fahssi, Mar 31 2008
Diagonal of rational function 1/(1 - (x^2 + 5*x*y + y^2)). - Gheorghe Coserea, Aug 01 2018
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LINKS
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FORMULA
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G.f.: 1/sqrt(1-10*x+21*x^2).
E.g.f.: exp(5x)*BesselI(0, 2x).
a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n,k)*binomial(2k,k). - Paul Barry, Mar 08 2005
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) = Sum_{k=0..n} 7^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 5*x + 26*x^2 + 140*x^3 + 777*x^4 + ... is the o.g.f. of A182401.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
a(n) = (1/Pi) * Integral_{x = -1..1} (3 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (7 - 4*x^2)^n/sqrt(1 - x^2) dx. (End)
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MATHEMATICA
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Table[SeriesCoefficient[1/(Sqrt[1-3*x]*Sqrt[1-7*x]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 15 2012 *)
CoefficientList[Series[1/(Sqrt[1-3x] Sqrt[1-7x]), {x, 0, 30}], x] (* Harvey P. Dale, Jun 20 2015 *)
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PROG
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(Maxima) a(n):=coeff(expand((1+5*x+x^2)^n), x^n);
(PARI) x='x+O('x^66); Vec(1/(sqrt(1-3*x)*sqrt(1-7*x))) \\ Joerg Arndt, May 11 2013
(PARI) {a(n) = sum(k=0, n, 7^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019
(PARI) {a(n) = sum(k=0, n\2, 5^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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