A020926 Expansion of 1/(1-4*x)^(15/2).
1, 30, 510, 6460, 67830, 624036, 5200300, 40116600, 290845350, 2003601300, 13223768580, 84151254600, 518932736700, 3113596420200, 18236779032600, 104557533120240, 588136123801350, 3252046802195700, 17705588145287700, 95051052148386600, 503770576386448980
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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GAP
List([0..20], n-> Binomial(2*(n+7), n+7)*Binomial(n+7,7)/3432); # G. C. Greubel, Jul 21 2019
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Magma
[&*[2*n+i: i in [1..13 by 2]]*Binomial(2*n, n)/135135: n in [0..20]]; // Vincenzo Librandi, Jul 05 2013
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Mathematica
CoefficientList[Series[1/(1-4x)^(15/2), {x,0,20}], x] (* Vincenzo Librandi, Jul 05 2013 *) -
PARI
vector(20, n, n--; m=n+7; binomial(2*m,m)*binomial(m,7)/3432) \\ G. C. Greubel, Jul 21 2019
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Sage
[binomial(2*(n+7), n+7)*binomial(n+7,7)/3432 for n in (0..20)] # G. C. Greubel, Jul 21 2019
Formula
a(n) = binomial(n+7, 7)*A000984(n+7)/A000984(7), A000984: central binomial coefficients. - Wolfdieter Lang
a(n) = ((2*n+13)*(2*n+11)*(2*n+9)*(2*n+7)*(2*n+5)*(2*n+3)*(2*n+1)/135135) * binomial(2*n, n). - Vincenzo Librandi, Jul 05 2013
Boas-Buck recurrence: a(n) = (30/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+7, 7). See a comment there. - Wolfdieter Lang, Aug 10 2017
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 2106*sqrt(3)*Pi - 13234624/1155.
Sum_{n>=0} (-1)^n/a(n) = 162500*sqrt(5)*log(phi) - 121172896/693, where phi is the golden ratio (A001622). (End)