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

0,1

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).

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

a(n) = A000292(n+1)*A000292(n+5). - 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)

Example

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.

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 *)

Prog

(PARI) vector(30, n, n--; binomial(n+3, 3)*binomial(n+7, 3)) \\ Michel Marcus, Jul 31 2015
(Magma) [Binomial(n+3, 3)*Binomial(n+7, 3): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015
(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

Crossrefs

Cf. A000292, A000580, A062264.

Sequence in context: A300523 A257758 A195968 * A171471 A223648 A219474

Adjacent sequences: A104471 A104472 A104473 * A104475 A104476 A104477

Keyword

easy,nonn

Author

Zerinvary Lajos, Apr 18 2005

Extensions

More terms from Robert G. Wilson v, Apr 20 2005

Status

approved