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A186492 Recursive triangle for calculating A186491. 4
1, 0, 1, 2, 0, 3, 0, 14, 0, 15, 28, 0, 132, 0, 105, 0, 5556, 0, 1500, 0, 945, 1112, 0, 10668, 0, 1995, 0, 10395, 0, 43784, 0, 212940, 0, 304290, 0, 135135, 87568, 0, 1408992, 0, 4533480, 0, 5239080, 0, 2027025 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The table entries are defined by a recurrence relation (see below).

This triangle can be used to calculate the entries of A186491: the nonzero entries of the first column of the triangle give A186491.

PRODUCTION MATRIX

The production matrix P for this triangle is the bidiagonal matrix with the sequence [2,4,6,...] on the main subdiagonal, the sequence [1,3,5,...] on the main superdiagonal and 0's elsewhere: the first row of P^n is the n-th row of this triangle.

LINKS

Table of n, a(n) for n=0..44.

C. V. Sukumar and A. Hodges, Quantum algebras and parity-dependent spectra, Proc. R. Soc. A (2007) 463, 2415-2427.

FORMULA

Recurrence relation

(1)... T(n,k) = (2*k-1)*T(n-1,k-1)+(2*k+2)*T(n-1,k+1).

GENERATING FUNCTION

E.g.f. (Compare with the e.g.f. of A104035):

(2)... 1/sqrt(cos(2*t)-u*sin(2*t)) = sum {n = 0..inf } R(n,u)*t^n/n! = 1 + u*t + (2+3*u^2)*t^2/2! + (14*u+15*u^3)*t^3/3!+....

ROW POLYNOMIALS

The row polynomials R(n,u) begin

... R(1,u) = u

... R(2,u) = 2+3*u^2

... R(3,u) = 14*u+15*u^3

... R(4,u) = 28+132*u^2+105u^4.

They satisfy the recurrence relation

(3)... R(n+1,u) = 2*(1+u^2)*d/du(R(n,u))+u*R(n,u) with starting value R(0,u) = 1.

Compare with Formula (1) of A104035 for the polynomials Q_n(u).

The polynomials R(n,u) are related to the shifted row polynomials A(n,u) of A142459 via

(4)... R(n,u) = ((u+I)/2)^n*A(n+1,(u-I)/(u+I))

with the inverse identity

(5)... A(n+1,u) = (-I)^n*(1-u)^n*R(n,I*(1+u)/(1-u)),

where {A(n,u)}n>=1 begins [1,1+u,1+10*u+u^2,1+59*u+59*u^2+u^3,...] and I = sqrt(-1).

EXAMPLE

Table begins

n\k|.....0.....1......2.....3......4.....5......6

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

0..|.....1

1..|.....0.....1

2..|.....2.....0......3

3..|.....0....14......0....15

4..|....28.....0....132.....0....105

5..|.....0...556......0..1500......0...945

6..|..1112.....0..10668.....0..19950.....0..10395

..

Examples of recurrence relation

T(4,2) = 3*T(3,1) + 6*T(3,3) = 3*14 + 6*15 = 132;

T(6,4) = 7*T(5,3) + 10*T(5,5) = 7*1500 + 10*945 = 19950.

MATHEMATICA

R[0][_] = 1; R[1][u_] = u;

R[n_][u_] := R[n][u] = 2(1+u^2) R[n-1]'[u] + u R[n-1][u];

Table[CoefficientList[R[n][u], u], {n, 0, 8}] // Flatten (* Jean-Fran├žois Alcover, Nov 13 2019 *)

CROSSREFS

A104035, A142459, A144015 (row sums), A186491.

Sequence in context: A079981 A117776 A298610 * A137448 A240606 A324379

Adjacent sequences:  A186489 A186490 A186491 * A186493 A186494 A186495

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Feb 22 2011

STATUS

approved

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Last modified December 11 21:00 EST 2019. Contains 329937 sequences. (Running on oeis4.)