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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). 11
1, 1, 1, 5, 6, 1, 61, 75, 15, 1, 1385, 1708, 350, 28, 1, 50521, 62325, 12810, 1050, 45, 1, 2702765, 3334386, 685575, 56364, 2475, 66, 1, 199360981, 245951615, 50571521, 4159155, 183183, 5005, 91, 1, 19391512145, 23923317720, 4919032300, 404572168 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

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

LINKS

Table of n, a(n) for n=0..39.

W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

FORMULA

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.

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.

Sum_{k=0..n} (-1)^k*T(n, k)*4^(n-k)= A000281(n) . - Philippe Deléham, Jan 26 2004

Sum_{k, 0<=k<=n} T(n, k)*(-4)^k*9^(n-k) = A002438(n+1) . - Philippe Deléham, Aug 26 2005

Sum_{k, 0<=k<=n}(-1)^k*9^(n-k)*T(n,k)=A000436(n) . - Philippe Deléham, Oct 27 2006

From Peter Bala, Aug 06 2013: (Start)

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).

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.

It appears that for arbitrary complex x we have lim {n -> inf} 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.

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

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

EXAMPLE

1;

1, 1;

5, 6, 1;

61, 75, 15, 1;

1385, 1708, 350, 28, 1;

From Peter Bala, Aug 06 2013: (Start)

Polynomial  |        Real zeros to 5 decimal places

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

R(5,-x)     | 1, 9.18062, 13.91597

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

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

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

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

(End)

MAPLE

A086646 := proc(n, k)

    if k < 0 or k > n then

        0 ;

    else

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

    end if;

end proc: # R. J. Mathar, Mar 14 2013

CROSSREFS

Cf. A000364 A005647 A084938.

Cf. A000281.

Row sums : A000795.

A055133, A086645 (unsigned matrix inverse), A103364, A104033.

Sequence in context: A105577 A054655 A086745 * A181612 A216808 A113106

Adjacent sequences:  A086643 A086644 A086645 * A086647 A086648 A086649

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jul 26 2003

STATUS

approved

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Last modified December 4 21:51 EST 2016. Contains 278755 sequences.