A169937 a(n) = binomial(m+n-1,n)^2 - binomial(m+n,n+1)*binomial(m+n-2,n-1) with m = 14.

1, 91, 3185, 63700, 866320, 8836464, 71954064, 488259720, 2848181700, 14620666060, 67255063876, 281248448936, 1081724803600, 3863302870000, 12914469594000, 40680579221100, 121443493851225, 345280521733875, 938920716995625, 2451077240157000, 6162708489537600

Offset: 0

Author

N. J. A. Sloane, Aug 28 2010

Keywords

• nonn
• easy

Comments

13th column (and diagonal) of the triangle A001263. - Bruno Berselli, May 07 2012

References

• S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=14.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

Crossrefs

The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces the sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.

Cf. A002378.

Programs

Magma

[(1/13)*Binomial(n+12,12)^2*(n+13)/(n+1): n in [0..20]]; // Bruno Berselli, Nov 09 2011

Maple

f:=m->[seq( binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1), n=0..20)]; f(14);

Mathematica

Table[Binomial[13+n,n]^2-Binomial[14+n,n+1]Binomial[12+n,n-1],{n,0,20}] (* Harvey P. Dale, Nov 09 2011 *)

PARI

a(n)=binomial(n+12,12)^2*(n+13)/(n+1)/13 \\ Charles R Greathouse IV, Nov 09 2011

Formula

a(n) = (1/13)*A010965(n+12)^2*(n+13)/(n+1). - Bruno Berselli, Nov 09 2011

a(n) = Product_{i=1..12} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016

From Amiram Eldar, Oct 19 2020: (Start)

Sum_{n>=0} 1/a(n) = 45997360927193/23100 - 201753552*Pi^2.

Sum_{n>=0} (-1)^n/a(n) = 16431564019/23100 - 72072*Pi^2. (End)