A053132 One half of binomial coefficients C(2*n-4,5).

3, 28, 126, 396, 1001, 2184, 4284, 7752, 13167, 21252, 32890, 49140, 71253, 100688, 139128, 188496, 250971, 329004, 425334, 543004, 685377, 856152, 1059380, 1299480, 1581255, 1909908, 2291058, 2730756, 3235501, 3812256, 4468464

Offset: 5

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 5..200
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000217, A053127.

Programs

Haskell

a053132 n = a053132_list !! (n-5)
a053132_list = f [1] $ drop 2 a000217_list where
   f xs ts'@(t:ts) = (sum $ zipWith (*) xs ts') : f (t:xs) ts
-- Reinhard Zumkeller, Mar 03 2015

Magma

[Binomial(2*n-4,5)/2: n in [5..40]]; // Vincenzo Librandi, Oct 07 2011

Mathematica

Binomial[2*Range[5,40]-4,5]/2 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{3,28,126,396,1001,2184},40] (* Harvey P. Dale, Oct 25 2015 *)

PARI

for(n=5,50, print1(binomial(2*n-4,5)/2, ", ")) \\ G. C. Greubel, Aug 26 2018

Formula

a(n) = binomial(2*n-4, 5)/2 if n >= 5 else 0.

G.f.: (x^5)*(3+10*x+3*x^2)/(1-x)^6.

a(n) = A053127(n)/2

a(n) = Sum_{k=1..n-4} (A000217(k)*A000217(2*n-k-7)). - Reinhard Zumkeller, Mar 03 2015

From Amiram Eldar, Jan 10 2022: (Start)

Sum_{n>=5} 1/a(n) = 335/6 - 80*log(2).

Sum_{n>=5} (-1)^(n+1)/a(n) = 85/6 - 20*log(2). (End)