A104474 a(n) = binomial(n+3,3)*binomial(n+7,3).
35, 224, 840, 2400, 5775, 12320, 24024, 43680, 75075, 123200, 194480, 297024, 440895, 638400, 904400, 1256640, 1716099, 2307360, 3059000, 4004000, 5180175, 6630624, 8404200, 10556000, 13147875, 16248960, 19936224, 24295040, 29419775
Offset: 0
Examples
a(0): C(0+3,3)*C(0+7,3) = C(3,3)*C(7,3) = 1*35 = 35. a(7): C(7+3,3)*C(7+7,3) = C(10,3)*(14,3) = 120*364 = 43680.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Programs
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Magma
[Binomial(n+3,3)*Binomial(n+7,3): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015
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Mathematica
f[n_] := Binomial[n + 3, 3]Binomial[n + 7, 3]; Table[ f[n], {n, 0, 28}] (* Robert G. Wilson v, Apr 20 2005 *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{35,224,840,2400,5775,12320,24024},40] (* Harvey P. Dale, May 25 2025 *) -
PARI
vector(30, n, n--; binomial(n+3,3)*binomial(n+7,3)) \\ Michel Marcus, Jul 31 2015
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SageMath
def A104474(n): return 140*binomial(n+7,7)//(n+4) print([A104474(n) for n in range(31)]) # G. C. Greubel, Mar 05 2025
Formula
a(n) = (1/36)*(n+1)*(n+2)*(n+3)*(n+5)*(n+6)*(n+7).
G.f.: (35 - 21*x + 7*x^2 - x^3)/(1-x)^7. - R. J. Mathar, Nov 30 2015
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=0} 1/a(n) = 7/200.
Sum_{n>=0} (-1)^n/a(n) = 1/40. (End)
From G. C. Greubel, Mar 05 2025: (Start)
a(n) = 140*A000580(n+7)/(n+4).
E.g.f.: (1/36)*(1260 + 6804*x + 7686*x^2 + 3102*x^3 + 531*x^4 + 39*x^5 + x^6)*exp(x). (End)
Extensions
More terms from Robert G. Wilson v, Apr 20 2005