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A269944 Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.

%I A269944 #50 Aug 03 2024 05:17:22
%S A269944 1,0,1,0,1,1,0,4,5,1,0,36,49,14,1,0,576,820,273,30,1,0,14400,21076,
%T A269944 7645,1023,55,1,0,518400,773136,296296,44473,3003,91,1,0,25401600,
%U A269944 38402064,15291640,2475473,191620,7462,140,1
%N A269944 Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.
%C A269944 Also known as central factorial numbers |t(2*n, 2*k)| (cf. A008955).
%C A269944 The analog for the Stirling set numbers is A269945.
%H A269944 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.
%H A269944 Peter Luschny, <a href="https://github.com/PeterLuschny/PartitionTransform/blob/main/PartitionTransform.ipynb">The Partition Transform -- A SageMath Jupyter Notebook</a>.
%H A269944 B. K. Miceli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Miceli/miceli4.html">Two q-Analogues of Poly-Stirling Numbers</a>, J. Integer Seq., 14 (2011), 11.9.6.
%F A269944 T(n,k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = (n - 1)^2 / (n*(4*n - 2)). The P-transform is defined in the link. See the Sage and Maple implementations below.
%F A269944 T(n, 1) = ((n - 1)!)^2 for n >= 1 (cf. A001044).
%F A269944 T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
%F A269944 Row sums: Product_{k=1..n} ((k - 1)^2 + 1) for n >= 0 (cf. A101686).
%F A269944 From _Fabián Pereyra_, Apr 25 2022: (Start)
%F A269944 T(n,k) = (-1)^(n-k)*Sum_{j=2*k..2*n} Stirling1(2*n,j)*binomial(j,2*k)*(n-1)^(j-2*k).
%F A269944 T(n,k) = Sum_{j=0..2*k} (-1)^(j - k)*Stirling1(n, j)*Stirling1(n, 2*k - j). (End)
%F A269944 From _Peter Luschny_, Feb 29 2024: (Start)
%F A269944 T(n, k) = (-1)^k*[x^(2*k)] P(x, n) where P(x, n) = Product_{j=0..n-1} (j-x)*(j+x).
%F A269944 T(n, k) = (2*n)!*[t^(n-k)] [x^(2*n)] cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)). (End)
%F A269944 T(n, k) = (-1)^k*[x^k] Pochhammer(-sqrt(x), n) * Pochhammer(sqrt(x), n). - _Peter Luschny_, Aug 03 2024
%e A269944 Triangle starts:
%e A269944   [1]
%e A269944   [0,     1]
%e A269944   [0,     1,     1]
%e A269944   [0,     4,     5,    1]
%e A269944   [0,    36,    49,   14,    1]
%e A269944   [0,   576,   820,  273,   30,  1]
%e A269944   [0, 14400, 21076, 7645, 1023, 55, 1]
%p A269944 T := proc(n, k) option remember; if n=k then return 1 fi; if k<0 or k>n then return 0 fi; T(n-1, k-1)+(n-1)^2*T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..8);
%p A269944 # Alternatively with the P-transform (cf. A269941):
%p A269944 A269944_row := n -> PTrans(n, n->`if`(n=1, 1, (n-1)^2/(n*(4*n-2))), (n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269944_row(n)), n=0..8);
%p A269944 # From _Peter Luschny_, Feb 29 2024: (Start)
%p A269944 # Computed as the coefficients of polynomials:
%p A269944 P := (x, n) -> local j; mul((j - x)*(j + x), j = 0..n-1):
%p A269944 T := (n, k) -> (-1)^k*coeff(P(x, n), x, 2*k):
%p A269944 for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
%p A269944 # Alternative, using the exponential generating function:
%p A269944 egf := cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)):
%p A269944 ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
%p A269944 Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, n-k), k = 0..n):
%p A269944 seq(print(Trow(n)), n = 0..9);  # (End)
%p A269944 # Alternative, row polynomials:
%p A269944 rowpoly := n -> pochhammer(-sqrt(x), n) * pochhammer(sqrt(x), n):
%p A269944 row := n -> local k; seq((-1)^k*coeff(expand(rowpoly(n)), x, k), k = 0..n):
%p A269944 seq(print(row(n)), n = 0..6);  # _Peter Luschny_, Aug 03 2024
%t A269944 T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + (n - 1)^2*T[n - 1, k]; T[_, _] = 0; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%t A269944 (* _Jean-François Alcover_, Jul 25 2019 *)
%o A269944 (Sage)
%o A269944 stircycle2 = lambda n: 1 if n == 1 else (n-1)^2/(n*(4*n-2))
%o A269944 norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
%o A269944 M = PtransMatrix(7, stircycle2, norm)
%o A269944 for m in M: print(m)
%Y A269944 Variants: A204579 (signed, row 0 missing), A008955.
%Y A269944 Cf. A007318 (order 0), A132393 (order 1), A269947 (order 3).
%Y A269944 Cf. A000330 (subdiagonal), A001044 (column 1), A101686 (row sums), A269945 (Stirling set), A269941 (P-transform).
%K A269944 nonn,tabl
%O A269944 0,8
%A A269944 _Peter Luschny_, Mar 22 2016