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%I #60 Feb 18 2024 03:33:34
%S 1,66,1716,26026,273273,2186184,14158144,77364144,367479684,
%T 1551580888,5924217936,20734762776,67255063876,203982391536,
%U 582806832960,1578435172600,4073755482225,10064572367850,23896002230100,54709268263650,121141951155225,260114319363600
%N a(n) = binomial(m+n-1,n)^2 - binomial(m+n,n+1)*binomial(m+n-2,n-1) with m=12.
%C 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.
%C 11th column (and diagonal) of the triangle A001263. - _Bruno Berselli_, May 07 2012
%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=12. - _N. J. A. Sloane_, Aug 28 2010.
%H Bruno Berselli, <a href="/A140925/b140925.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).
%F (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)).
%F 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
%F a(n) = ((n+11)/(11*n+11))*binomial(n+10,10)^2. - _Bruno Berselli_, May 07 2012
%F a(n) = Product_{i=1..10} A002378(n+i)/A002378(i). - _Bruno Berselli_, Sep 01 2016
%F From _Amiram Eldar_, Oct 19 2020: (Start)
%F Sum_{n>=0} 1/a(n) = 186224135603/2352 - 8022300*Pi^2.
%F Sum_{n>=0} (-1)^n/a(n) = 11550*Pi^2 - 114904889/1008. (End)
%t Table[Binomial[11 + n, n]^2 - Binomial[12 + n, n + 1] Binomial[10 + n, n - 1], {n, 0, 21}] (* _Bruno Berselli_, May 07 2012 *)
%o (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 */
%o (Magma) [((n+11)/(11*n+11))*Binomial(n+10,10)^2: n in [0..21]]; // _Bruno Berselli_, May 07 2012
%o (PARI) a(n) = ((n/11+1)/(n+1))*binomial(n+10,10)^2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A001263, A002378, A134291, A140934.
%K nonn,easy
%O 0,2
%A _R. H. Hardin_, Jul 05 2008
%E Edited by _N. J. A. Sloane_, Aug 28 2010
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