A171128 A117852*A130595 as lower triangular matrices.

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 19, 28, 18, 4, 1, 51, 95, 70, 30, 5, 1, 141, 306, 285, 140, 45, 6, 1, 393, 987, 1071, 665, 245, 63, 7, 1, 1107, 3144, 3948, 2856, 1330, 392, 84, 8, 1, 3139, 9963, 14148, 11844, 6426, 2394, 588, 108, 9, 1

Offset

0,4

Comments

Mirror image of triangle in A135091.

Exponential Riordan array [exp(x)*Bessel_I(0,2*x), x] = A007318 * A109187. - Peter Bala, Feb 12 2017

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Formula

Sum_{k=0..n} T(n,k)*x^k = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n), A104454(n) for x = -1,0,1,2,3,4,5,6 respectively.

T(n,k) = binomial(n,k)*A002426(n-k). - Philippe Deléham, Dec 12 2009

From Peter Bala, Feb 12 2017: (Start)

T(n,k) = Sum_{j = 0..floor((n-k)/2)} n!/((n-k-2*j)!*j!^2*k!).

T(n,k) = n/k*T(n-1,k-1) with T(n,0) = A002426(n).

(n - k)^2*T(n,k) = n*(2*n - 2*k - 1)*T(n-1,k) + 3*n*(n - 1)*T(n-2,k).

O.g.f. = 1/sqrt((1 - (1 + t)*z)^2 - 4*z^2) = 1 + (1 + t)*z + (3 + 2*t + t^2)*z^2 + (7 + 9*t + 3*t^2 + t^3 )*z^3 + ....

E.g.f. Bessel_I(0,2*x) * exp((1 + t)*x) = 1 + (1 + t)*z + (3 + 2*t + t^2)*z^2/2! + (7 + 9*t + 3*t^2 + t^3 )*z^3/3! + ....

n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k)*binomial(2*k,k)*(1 + t)^(n-2*k) = coefficient of x^n in the expansion of (1 + (1 + t)*x + x^2)^n.

The polynomials R(n, t - 1) are the row polynomials of A109187.

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

Moment representation on a finite interval: R(n,t) = 1/Pi * Integral_{x = t-1 .. t+3} x^n/sqrt((t + 3 - x)*(x - t + 1)) dx.

The zeros of the row polynomials appear to lie on the vertical line Re(z) = -1 in the complex plane, and the zeros of R(n,t) and R(n+1,t) appear to interlace along this line.

(End)

Example

Triangle begins:
   1
   1  1
   3  2  1
   7  9  3 1
  19 28 18 4 1
  ...
From Peter Bala, Feb 12 2017: (Start)
The infinitesimal generator begins
      0
      1    0
      2    2     0
      0    6     3     0
     -6    0    12     4     0
      0  -30     0    20     5   0
     80    0   -90     0    30   6   0
      0  560     0  -210     0  42   7  0
  -2310    0  2240     0  -420   0  56  8  0
  ....
and equals the generalized exponential Riordan array [x + log(Bessel_I(0,2*x), x], and so has integer entries. (End)

Mathematica

A002426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}]; Table[ Binomial[n, k]*A002426[n - k], {n, 0, 99}, {k, 0, n} ]//Flatten (* G_. C. Greubel_, Mar 07 2017 *)

Crossrefs

A000984 (row sums), A135091 (row reversed). Cf. A002426, A117852, A130595, A109187.

Sequence in context: A359413 A192020 A367564 * A122832 A056151 A134436

Adjacent sequences: A171125 A171126 A171127 * A171129 A171130 A171131

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 04 2009

Status

approved