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A006333 From the enumeration of corners.

%I A006333 M2175 #31 Aug 02 2024 22:38:29
%S A006333 0,2,60,660,4290,20020,74256,232560,639540,1586310,3617900,7696260,
%T A006333 15438150,29451240,53796160,94607040,160908264,265670730,427156860,
%U A006333 670609940,1030350090,1552346268,2297341200,3344614000,4796473500
%N A006333 From the enumeration of corners.
%D A006333 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006333 Matthew House, <a href="/A006333/b006333.txt">Table of n, a(n) for n = 0..10000</a>
%H A006333 G. Kreweras, <a href="http://www.numdam.org/numdam-bin/item?id=BURO_1965__6__9_0">Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
%H A006333 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A006333 a(n) = (n*(1 + n)^2*(2 + n)^2*(3 + n)*(1 + 2*n)*(3 + 2*n)*(5 + 2*n))/7560.
%F A006333 G.f.: 2*(1 + 20*x + 75*x^2 + 75*x^3 + 20*x^4 + x^5)/(1-x)^10.
%t A006333 Abs@ With[{n = 4}, Table[(2 (-1)^(n + k) (n + k - 1)! (2 n + 2 k - 3)!)/(n! k! (2 n - 1)! (2 k - 1)!), {k, 0, 24}]] (* or *)
%t A006333 {0}~Join~CoefficientList[Series[2 (1 + 20 x + 75 x^2 + 75 x^3 + 20 x^4 + x^5)/(1 - x)^10, {x, 0, 23}], x] (* _Michael De Vlieger_, Mar 26 2016 *)
%t A006333 LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,2,60,660,4290,20020,74256,232560,639540,1586310},30] (* _Harvey P. Dale_, Jan 01 2017 *)
%o A006333 (PARI) a(n) = (n*(1 + n)^2*(2 + n)^2*(3 + n)*(1 + 2*n)*(3 + 2*n)*(5 + 2*n))/7560 \\ _Charles R Greathouse IV_, Jul 28 2015
%o A006333 (PARI) x='x+O('x^99); concat(0, Vec(2*(1+20*x+75*x^2+75*x^3+20*x^4+x^5)/(1-x)^10)) \\ _Altug Alkan_, Mar 26 2016
%Y A006333 A row of A132339.
%K A006333 nonn,easy
%O A006333 0,2
%A A006333 _N. J. A. Sloane_