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A086646 Triangle, read by rows, of numbers T(n,k), 0 <= k <= n, given by T(n,k) = A000364(n-k)*binomial(2*n, 2*k).

%I A086646 #59 Jan 21 2025 23:22:21
%S A086646 1,1,1,5,6,1,61,75,15,1,1385,1708,350,28,1,50521,62325,12810,1050,45,
%T A086646 1,2702765,3334386,685575,56364,2475,66,1,199360981,245951615,
%U A086646 50571521,4159155,183183,5005,91,1,19391512145,23923317720,4919032300,404572168,17824950,488488,9100,120,1
%N A086646 Triangle, read by rows, of numbers T(n,k), 0 <= k <= n, given by T(n,k) = A000364(n-k)*binomial(2*n, 2*k).
%C A086646 The elements of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^(n+k)*A086645(n,k). - _R. J. Mathar_, Mar 14 2013
%C A086646 Let E(y) = Sum_{n >= 0} y^n/(2*n)! = cosh(sqrt(y)). Then this triangle is the generalized Riordan array (1/E(-y), y) with respect to the sequence (2*n)! as defined in Wang and Wang. - _Peter Bala_, Aug 06 2013
%C A086646 Let P_n be the poset of even size subsets of [2n] ordered by inclusion.  Then Sum_{k=0..n}(-1)^(n-k)*T(n,k)*x^k is the characteristic polynomial of P_n. - _Geoffrey Critzer_, Feb 24 2021
%H A086646 Alois P. Heinz, <a href="/A086646/b086646.txt">Rows n = 0..140, flattened</a>
%H A086646 Tom Copeland, <a href="https://tcjpn.wordpress.com/2020/07/11/skipping-over-dimensions-juggling-zeros-in-the-matrix/">Skipping over Dimensions, Juggling Zeros in the Matrix</a>, 2020.
%H A086646 W. Wang and T. Wang, <a href="https://doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
%F A086646 cosh(u*t)/cos(t) = Sum_{n>=0} S_2n(u)*(t^(2*n))*(1/(2*n)!). S_2n(u) = Sum_{k>=0} T(n,k)*u^(2*k). Sum_{k>=0} (-1)^k*T(n,k) = 0. Sum_{k>=0} T(n,k) = 2^n*A005647(n); A005647: Salie numbers.
%F A086646 Triangle T(n,k) read by rows; given by [1, 4, 9, 16, 25, 36, 49, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.
%F A086646 Sum_{k=0..n} (-1)^k*T(n,k)*4^(n-k) = A000281(n). - _Philippe Deléham_, Jan 26 2004
%F A086646 Sum_{k=0..n} T(n,k)*(-4)^k*9^(n-k) = A002438(n+1). - _Philippe Deléham_, Aug 26 2005
%F A086646 Sum_{k=0..n} (-1)^k*9^(n-k)*T(n,k) = A000436(n). - _Philippe Deléham_, Oct 27 2006
%F A086646 From _Peter Bala_, Aug 06 2013: (Start)
%F A086646 Let E(y) = Sum_{n >= 0} y^n/(2*n)! = cosh(sqrt(y)). Generating function: E(x*y)/E(-y) = 1 + (1 + x)*y/2! + (5 + 6*x + x^2)*y^2/4! + (61 + 75*x + 15*x^2 + x^3)*y^3/6! + .... The n-th power of this array has a generating function E(x*y)/E(-y)^n. In particular, the matrix inverse is a signed version of A086645 with a generating function E(-y)*E(x*y).
%F A086646 Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} (-1)^(n-k)*binomial(2*n,2*k)*R(k,x) with initial value R(0,x) = 1.
%F A086646 It appears that for arbitrary complex x we have lim_{n -> infinity} R(n,-x^2)/R(n,0) = cos(x*Pi/2). A stronger result than pointwise convergence may hold: the convergence may be uniform on compact subsets of the complex plane. This would explain the observation that the real zeros of the polynomials R(n,-x) seem to converge to the odd squares 1, 9, 25, ... as n increases. Some numerical examples are given below. Cf. A055133, A091042 and A103364.
%F A086646 R(n,-1) = 0; R(n,-9) = (-1)^n*2*4^n; R(n,-25) = (-1)^n*2*(16^n - 4^n);
%F A086646 R(n,-49) = (-1)^n*2*(36^n - 16^n + 4^n). (End)
%e A086646 Triangle begins:
%e A086646       1;
%e A086646       1,     1;
%e A086646       5,     6,     1;
%e A086646      61,    75,    15,    1;
%e A086646    1385,  1708,   350,   28,  1;
%e A086646   50521, 62325, 12810, 1050, 45, 1;
%e A086646   ...
%e A086646 From _Peter Bala_, Aug 06 2013: (Start)
%e A086646 Polynomial  |        Real zeros to 5 decimal places
%e A086646 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A086646 R(5,-x)     | 1, 9.18062, 13.91597
%e A086646 R(10,-x)    | 1, 9.00000, 25.03855,  37.95073
%e A086646 R(15,-x)    | 1, 9.00000, 25.00000,  49.00895, 71.83657
%e A086646 R(20,-x)    | 1, 9.00000, 25.00000,  49.00000, 81.00205, 114.87399
%e A086646 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A086646 (End)
%p A086646 A086646 := proc(n,k)
%p A086646     if k < 0 or k > n then
%p A086646         0 ;
%p A086646     else
%p A086646         A000364(n-k)*binomial(2*n,2*k) ;
%p A086646     end if;
%p A086646 end proc: # _R. J. Mathar_, Mar 14 2013
%t A086646 R[0, _] = 1;
%t A086646 R[n_, x_] := R[n, x] = x^n - Sum[(-1)^(n-k) Binomial[2n, 2k] R[k, x], {k, 0, n-1}];
%t A086646 Table[CoefficientList[R[n, x], x], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Dec 19 2019 *)
%t A086646 T[0, 0] := 1; T[n_, 0] := -Sum[(-1)^k T[n, k], {k, 1, n}]; T[n_, k_]/;0<k<=n := T[n-1, k-1] (n(2n-1))/(k(2k-1)); T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* _Oliver Seipel_, Jan 11 2025 *)
%Y A086646 Cf. A000364, A005647, A084938.
%Y A086646 Cf. A000281.
%Y A086646 Cf. A000795 (row sums).
%Y A086646 Cf. A055133, A086645 (unsigned matrix inverse), A103364, A104033.
%Y A086646 T(2n,n) give |A214445(n)|.
%K A086646 easy,nonn,tabl
%O A086646 0,4
%A A086646 _Philippe Deléham_, Jul 26 2003