A053138 Binomial coefficients C(2*n+9,9).
1, 55, 715, 5005, 24310, 92378, 293930, 817190, 2042975, 4686825, 10015005, 20160075, 38567100, 70607460, 124403620, 211915132, 350343565, 563921995, 886163135, 1362649145, 2054455634, 3042312350, 4431613550, 6358402050, 8996462475, 12565671261, 17341763505
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Milan Janjić, Two Enumerative Functions.
- Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45, 10,-1).
Programs
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Magma
[Binomial(2*n+9,9): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011
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Maple
A053138:=n->binomial(2*n+9,9): seq(A053138(n), n=0..40); # Wesley Ivan Hurt, Dec 05 2016
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Mathematica
Table[Binomial[2n+9,9],{n,0,30}] (* Harvey P. Dale, Dec 08 2011 *) -
PARI
for(n=0,30, print1(binomial(2*n+9, 9), ", ")) \\ G. C. Greubel, Sep 03 2018
Formula
a(n) = binomial(2*n+9, 9) = A000582(2*n+9).
G.f.: (1 + 45*x + 210*x^2 + 210*x^3 + 45*x^4 + x^5) / (1-x)^10.
G.f.: (1 + x) * (x^4 + 44*x^3 + 166*x^2 + 44*x + 1) / (1-x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 9. - Wesley Ivan Hurt, Dec 05 2016
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=0} 1/a(n) = 1152*log(2) - 27912/35.
Sum_{n>=0} (-1)^n/a(n) = 36*Pi - 3924/35. (End)
Comments