A140925 a(n) = binomial(m+n-1,n)^2 - binomial(m+n,n+1)*binomial(m+n-2,n-1) with m=12.

1, 66, 1716, 26026, 273273, 2186184, 14158144, 77364144, 367479684, 1551580888, 5924217936, 20734762776, 67255063876, 203982391536, 582806832960, 1578435172600, 4073755482225, 10064572367850, 23896002230100, 54709268263650, 121141951155225, 260114319363600

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 triangle A001263. - 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

a(n) = Product_{i=1..10} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016

From Amiram 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

Cf. A001263, A002378, A134291, A140934.

Sequence in context: A133318 A197645 A270847 * A269779 A279446 A271797

Adjacent sequences: A140922 A140923 A140924 * A140926 A140927 A140928

Keyword

nonn,easy

Author

R. H. Hardin, Jul 05 2008

Extensions

Edited by N. J. A. Sloane, Aug 28 2010

Status

approved