OFFSET
0,2
COMMENTS
Number of 2 X 10 matrices with elements in 0..n with each row and each column in nondecreasing order. 2,10,n can be permuted, see formula.
11th column (and diagonal) of the triangleA001263.-Bruno Berselli,May 07 2012
REFERENCES
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=12. -N. J. A. Sloane,Aug 28 2010.
LINKS
Bruno Berselli,Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients,signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).
FORMULA
(Empirical) Set p,q,r to n,10,2 (in any order) in s=p+q+r-1; a(n) = product {i in 0..r-1} (binomial(s,p+i)*i!/(s-i)^(r-i-1)).
G.f.: (1 + x)*(1 + 44*x + 496*x^2 + 2024*x^3 + 3268*x^4 + 2024*x^5 + 496*x^6 + 44*x^7 + x^8)/(1 - x)^21. -Bruno Berselli,May 07 2012
a(n) = ((n+11)/(11*n+11))*binomial(n+10,10)^2. -Bruno Berselli,May 07 2012
FromAmiram Eldar,Oct 19 2020: (Start)
Sum_{n>=0} 1/a(n) = 186224135603/2352 - 8022300*Pi^2.
Sum_{n>=0} (-1)^n/a(n) = 11550*Pi^2 - 114904889/1008. (End)
MATHEMATICA
Table[Binomial[11 + n, n]^2 - Binomial[12 + n, n + 1] Binomial[10 + n, n - 1], {n, 0, 21}] (*Bruno Berselli,May 07 2012 *)
PROG
(Maxima) makelist(coeff(taylor((1+x)*(1+44*x+496*x^2+2024*x^3+3268*x^4+2024*x^5+496*x^6+44*x^7+x^8)/(1-x)^21, x, 0, n), x, n), n, 0, 21); /*Bruno Berselli,May 07 2012 */
(Magma) [((n+11)/(11*n+11))*Binomial(n+10, 10)^2: n in [0..21]]; //Bruno Berselli,May 07 2012
(PARI) a(n) = ((n/11+1)/(n+1))*binomial(n+10, 10)^2 \\Charles R Greathouse IV,Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin,Jul 05 2008
EXTENSIONS
Edited byN. J. A. Sloane,Aug 28 2010
STATUS
approved