A104475 - OEIS

%I #25 Mar 05 2025 03:13:36

%S 70,630,3150,11550,34650,90090,210210,450450,900900,1701700,3063060,

%T 5290740,8817900,14244300,22383900,34321980,51482970,75710250,

%U 109359250,155405250,217567350,300450150,409704750,552210750,736281000,971890920,1270934280,1647507400,2118223800

%N a(n) = binomial(n+4,4) * binomial(n+8,4).

%H G. C. Greubel, <a href="/A104475/b104475.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = 70*A000581(n-8). - _Michel Marcus_, Jul 31 2015

%F From _Amiram Eldar_, Aug 30 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 4/245.

%F Sum_{n>=0} (-1)^n/a(n) = 512*log(2)/35 - 37216/3675. (End)

%F From _G. C. Greubel_, Mar 05 2025: (Start)

%F G.f.: 70/(1-x)^9.

%F E.g.f.: (1/576)*(40320 + 322560*x + 564480*x^2 + 376320*x^3 + 117600*x^4 + 18816*x^5 + 1568*x^6 + 64*x^7 + x^8)*exp(x). (End)

%e a(0): C(0+4,4)*C(0+8,4) = C(4,4)*C(8,4) = 1*70 = 70.

%e a(7): C(5+4,4)*C(5+8,4) = C(9,4)*(13,4) = 126*715 = 90090.

%p A104475:=n->binomial(n+4,4)*binomial(n+8,4): seq(A104475(n), n=0..40); # _Wesley Ivan Hurt_, Jan 29 2017

%t f[n_] := Binomial[n + 4, 4]Binomial[n + 8, 4]; Table[ f[n], {n, 0, 25}] (* _Robert G. Wilson v_, Apr 20 2005 *)

%o (PARI) vector(30, n, n--; binomial(n+4,4)*binomial(n+8,4)) \\ _Michel Marcus_, Jul 31 2015

%o (Magma) [Binomial(n+4,4)*Binomial(n+8,4): n in [0..30]]; // _Vincenzo Librandi_, Jul 31 2015

%o (SageMath)

%o def A104475(n): return binomial(n+4,4)*binomial(n+8,4)

%o print([A104475(n) for n in range(31)]) # _G. C. Greubel_, Mar 05 2025

%Y Cf. A000581, A062264.

%K easy,nonn,changed

%O 0,1

%A _Zerinvary Lajos_, Apr 18 2005

%E More terms from _Robert G. Wilson v_, Apr 20 2005