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A156995 Triangle T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2, read by rows.

%I A156995 #14 Sep 08 2022 08:45:41
%S A156995 2,1,2,2,4,2,6,12,9,2,24,48,40,16,2,120,240,210,100,25,2,720,1440,
%T A156995 1296,672,210,36,2,5040,10080,9240,5040,1764,392,49,2,40320,80640,
%U A156995 74880,42240,15840,4032,672,64,2,362880,725760,680400,393120,154440,42768
%N A156995 Triangle T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2, read by rows.
%C A156995 For n>=1, o.g.f. of n-th row is a polynomial p(x,n) = Sum_{k=0..n} ( 2*n*(n-k)! * binomial(2*n-k, k)/(2*n-k)) * x^k. These polynomials are hit polynomials for the reduced ménage problem (Riordan 1958).
%D A156995 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 197-199
%H A156995 G. C. Greubel, <a href="/A156995/b156995.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A156995 T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2.
%e A156995 Triangle starts with:
%e A156995   n=0:     2;
%e A156995   n=1:     1,     2;
%e A156995   n=2:     2,     4,     2;
%e A156995   n=3:     6,    12,     9,     2;
%e A156995   n=4:    24,    48,    40,    16,     2;
%e A156995   n=5:   120,   240,   210,   100,    25,    2;
%e A156995   n=6:   720,  1440,  1296,   672,   210,   36,   2;
%e A156995   n=7:  5040, 10080,  9240,  5040,  1764,  392,  49,  2;
%e A156995   n=8: 40320, 80640, 74880, 42240, 15840, 4032, 672, 64, 2;
%e A156995   ...
%t A156995 T[n_, k_]:= If[n==0, 2, 2*n*Binomial[2*n-k, k]*(n-k)!/(2*n-k)];
%t A156995 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 14 2021 *)
%o A156995 (Magma)
%o A156995 A156995:= func< n,k | n eq 0 select 2 else 2*n*Factorial(n-k)*Binomial(2*n-k, k)/(2*n-k) >;
%o A156995 [A156995(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 14 2021
%o A156995 (Sage)
%o A156995 def A156995(n,k): return 2 if (k==n) else 2*n*factorial(n-k)*binomial(2*n-k,k)/(2*n-k)
%o A156995 flatten([[A156995(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 14 2021
%Y A156995 Row sums are A300484.
%K A156995 nonn,tabl
%O A156995 0,1
%A A156995 _Roger L. Bagula_, Feb 20 2009
%E A156995 Edited and changed T(0,0) = 2 (to make formula continuous and constant along the diagonal k = n) by _Max Alekseyev_, Mar 06 2018