

A086646


Triangle, read by rows, of numbers T(n; k), 0<=k<=n, given by T(n; k) = A000364(nk)*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
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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, 64666500.


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^(nk)= A000281(n) .  Philippe Deléham, Jan 26 2004
Sum_{k, 0<=k<=n} T(n, k)*(4)^k*9^(nk) = A002438(n+1) .  Philippe Deléham, Aug 26 2005
Sum_{k, 0<=k<=n}(1)^k*9^(nk)*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 nth 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..n1} (1)^(nk)*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(nk)*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: A290319 A321630 A086745 * A181612 A216808 A293107
Adjacent sequences: A086643 A086644 A086645 * A086647 A086648 A086649


KEYWORD

easy,nonn,tabl


AUTHOR

Philippe Deléham, Jul 26 2003


STATUS

approved



