A062145 Triangle read by rows. T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.

1, 1, 4, 1, 10, 10, 1, 18, 45, 20, 1, 28, 126, 140, 35, 1, 40, 280, 560, 350, 56, 1, 54, 540, 1680, 1890, 756, 84, 1, 70, 945, 4200, 7350, 5292, 1470, 120, 1, 88, 1540, 9240, 23100, 25872, 12936, 2640, 165, 1, 108

Offset

0,3

Comments

Coefficient triangle of certain polynomials N(3; m,x).

Links

Table of n, a(n) for n=0..46.

Formula

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=3) Laguerre triangle L(3; n+m, m) = A062137(n+m, m), n >= 0, is N(3; m, x)/(1-x)^(2*(m+2)), with the row polynomials N(3; m, x) := Sum_{k=0..m} a(m, k)*x^k.

N(3; m, x) := ((1-x)^(2*(m+2)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+4))); a(m, k) = [x^k]N(3; m, x).

N(3; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+3-j)!/((m+3)!*(m-j)!))*(x^(m-j))*(1-x)^j).

N(3; m, x)= x^m*(2*m+3)! * 2F1(-m, -m; -2*m-3; (x-1)/x)/((m+3)!*m!). [Jean-François Alcover, Sep 18 2013]

Example

From Zerinvary Lajos, Jan 02 2006: (Start)
As a square array:
   1,   1,    1,     1,     1,    1,    1,  1,  1, ...
   4,  10,   18,    28,    40,   54,   70, 88, ...
  10,  45,  126,   280,   540,  945, 1540, ...
  20, 140,  560,  1680,  4200, 9240, ...
  35, 350, 1890,  7350, 23100, ...
  56, 756, 5292, 25872, ...
  ... (End)

Mathematica

NN[3, m_, x_] := x^m*(2*m+3)!*Hypergeometric2F1[-m, -m, -2*m-3, (x-1)/x]/((m+3)!*m!); Table[CoefficientList[NN[3, m, x], x], {m, 0, 9}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
P[c_, n_, z_] := Sum[Binomial[n, k] Pochhammer[n - k + c, k] z^k / k!, {k, 0, n}];
CL[c_] := Table[CoefficientList[P[c, n, z], z], {n, 0, 5}] // TableForm
CL[4]  (* Peter Luschny, Feb 12 2024 *)

Crossrefs

Family of polynomials: A008459 (c=1), A132813 (c=2), A062196 (c=3), this sequence (c=4), A062264 (c=5), A062190 (c=6).

Cf. A000292.

Sequence in context: A213765 A349809 A182971 * A178216 A307529 A019213

Adjacent sequences: A062142 A062143 A062144 * A062146 A062147 A062148

Keyword

nonn,tabl

Author

Wolfdieter Lang, Jun 19 2001

Extensions

New name by Peter Luschny, Feb 12 2024

Status

approved