A282252 Exponential Riordan array [Bessel_I(0,2*x)^2, x].

1, 0, 1, 4, 0, 1, 0, 12, 0, 1, 36, 0, 24, 0, 1, 0, 180, 0, 40, 0, 1, 400, 0, 540, 0, 60, 0, 1, 0, 2800, 0, 1260, 0, 84, 0, 1, 4900, 0, 11200, 0, 2520, 0, 112, 0, 1, 0, 44100, 0, 33600, 0, 4536, 0, 144, 0, 1, 63504, 0, 220500, 0, 84000, 0, 7560, 0, 180, 0, 1

Offset: 0

Author

Peter Bala, Feb 12 2017

Keywords

• nonn
• tabl
• easy

Comments

Bessel_I(0,2*x) = Sum_{n >= 0} binomial(2*n,n)*x^(2*n)/(2*n)! is a modified Bessel function of the first kind.

Consider the infinite 2-dimensional square lattice Z x Z with an oriented self-loop at each vertex. Then the triangle entry T(n,k) equals the number of walks of length n from the origin to itself having k loops. An example is given below.

See A069466 for walks an infinite 2-dimensional square lattice without self-loops.

This is the square of triangle A109187, whose entries give the number of walks of length n from a vertex to itself having k loops on a 1-dimensional integer lattice with an oriented self-loop at each vertex.

A109187 is the exponential Riordan array [Bessel_I(0,2*x), x]. Note that Bessel_I(0,2*x)^2 = (Sum_{n >= 0} binomial(2*n,n)* x^(2*n)/(2*n)!)^2 = Sum_{n >= 0} binomial(2*n,n)^2*x^(2*n) /(2*n)!.

Examples

			The triangle begins
    1;
    0,   1;
    4,   0,   1;
    0,  12,   0,   1;
   36,   0,  24,   0,   1;
    0, 180,   0,  40,   0,   1;
  400,   0, 540,   0,  60,   0,   1;
  ...
T(3,1) = 12: on the square lattice, let L, R, U, D denote a left step, right step, up step and down step respectively. The 12 walks of length 3 containing a single loop are
    loop L R, loop R L, loop U D, loop D U,
    L loop R, R loop L, U loop D, D loop U,
    L R loop, R L loop, U D loop, D U loop.
The infinitesimal generator of this array has integer entries and begins
      0;
      0,    0;
      4,    0,    0;
      0,   12,    0,    0;
    -12,    0,   24,    0,    0;
      0,  -60,    0,   40,    0,    0;
    160,    0, -180,    0,   60,    0,    0;
      0, 1120,    0, -420,    0,   84,    0,    0;
  -4620,    0, 4480,    0, -840,    0,  112,    0,    0;
  ...
It is the generalized exponential Riordan array [ 2*log(Bessel_I(0,2*x)), x ].
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

A201805 gives row sums. Cf. A069466, A109187.

Programs

Maple

T := (n, k) -> (1/2)*binomial(n, k)*binomial(n-k, floor((1/2)*n-(1/2)*k))^2*(1+(-1)^(n-k)):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

Table[Binomial[n, k] Binomial[n - k, Floor[(n - k)/2]]^2*(1 + (-1)^(n - k))/2, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 12 2017 *)

PARI

for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(n-k,floor((n-k)/2))^2*(1 + (-1)^(n-k))/2, ", "))) \\ G. C. Greubel, Aug 16 2017

Formula

T(n,k) = binomial(n,k)*binomial(n-k,floor((n-k)/2))^2*(1 + (-1)^(n-k))/2.

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

T(n,k) = the coefficient of t^k in the expansion of (t + X + 1/X + Y + 1/Y)^n.

T(n,k) = 1/Pi^2 * Integral_{y = 0..Pi} Integral_{x = 0..Pi} ( t + 2*cos(x) + 2*cos(y) )^n dx dy.

E.g.f.: exp(x*t)*Bessel_I(0,2*x)^2 = 1 + t*x + (4 + t^2)*x^2/2! + (12*t + t^3)*x^3/3! + (36 + 24*t^2 + t^4)*x^4/4! + ....

The n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k)*binomial(2*k,k)^2 * t^(n-2*k).

Recurrence: n^2*R(n,t) = t*(3*n^2 - 3*n + 1)*R(n-1,t) + (16 - 3*t^2)*(n - 1)^2*R(n-2,t) + t*(t^2 - 16)*(n - 1)*(n - 2)*R(n-3,t) with R(0,t) = 1, R(1,t) = t and R(2,t) = 4 + t^2.

d/dt(R(n,t)) = n*R(n-1,t).

The zeros of the row polynomials appear to lie on the imaginary axis in the complex plane. Also, the zeros of R(n,t) and R(n+1,t) appear to interlace on the imaginary axis.