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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).
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%I #54 Apr 27 2023 10:19:01

%S 1,1,1,5,6,1,61,75,15,1,1385,1708,350,28,1,50521,62325,12810,1050,45,

%T 1,2702765,3334386,685575,56364,2475,66,1,199360981,245951615,

%U 50571521,4159155,183183,5005,91,1,19391512145,23923317720,4919032300,404572168,17824950,488488,9100,120,1

%N 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 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 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 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 Alois P. Heinz, <a href="/A086646/b086646.txt">Rows n = 0..140, flattened</a>

%H T. 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 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 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 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 Sum_{k=0..n} (-1)^k*T(n,k)*4^(n-k) = A000281(n). - _Philippe Deléham_, Jan 26 2004

%F Sum_{k=0..n} T(n,k)*(-4)^k*9^(n-k) = A002438(n+1). - _Philippe Deléham_, Aug 26 2005

%F Sum_{k=0..n} (-1)^k*9^(n-k)*T(n,k) = A000436(n). - _Philippe Deléham_, Oct 27 2006

%F From _Peter Bala_, Aug 06 2013: (Start)

%F 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 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 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 R(n,-1) = 0; R(n,-9) = (-1)^n*2*4^n; R(n,-25) = (-1)^n*2*(16^n - 4^n);

%F R(n,-49) = (-1)^n*2*(36^n - 16^n + 4^n). (End)

%e Triangle begins:

%e 1;

%e 1, 1;

%e 5, 6, 1;

%e 61, 75, 15, 1;

%e 1385, 1708, 350, 28, 1;

%e 50521, 62325, 12810, 1050, 45, 1;

%e ...

%e From _Peter Bala_, Aug 06 2013: (Start)

%e Polynomial | Real zeros to 5 decimal places

%e = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

%e R(5,-x) | 1, 9.18062, 13.91597

%e R(10,-x) | 1, 9.00000, 25.03855, 37.95073

%e R(15,-x) | 1, 9.00000, 25.00000, 49.00895, 71.83657

%e R(20,-x) | 1, 9.00000, 25.00000, 49.00000, 81.00205, 114.87399

%e = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

%e (End)

%p A086646 := proc(n,k)

%p if k < 0 or k > n then

%p 0 ;

%p else

%p A000364(n-k)*binomial(2*n,2*k) ;

%p end if;

%p end proc: # _R. J. Mathar_, Mar 14 2013

%t R[0, _] = 1;

%t R[n_, x_] := R[n, x] = x^n - Sum[(-1)^(n-k) Binomial[2n, 2k] R[k, x], {k, 0, n-1}];

%t Table[CoefficientList[R[n, x], x], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Dec 19 2019 *)

%Y Cf. A000364, A005647, A084938.

%Y Cf. A000281.

%Y Cf. A000795 (row sums).

%Y Cf. A055133, A086645 (unsigned matrix inverse), A103364, A104033.

%Y T(2n,n) give |A214445(n)|.

%K easy,nonn,tabl

%O 0,4

%A _Philippe Deléham_, Jul 26 2003