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A062145 Coefficient triangle of certain polynomials N(3; m,x). 19
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Zerinvary Lajos, Mar 24 2005: (Start)

Formatted as an upper right triangle:

C(0,0)*C(3,0),C(1,1)*C(4,0),C(2,2)*C(5,0),C(3,3)*C(6,0), C(4,4)*C(7,0),C(5,5)*C(8,0),C(6,6)*C(9,0),C(7,7)*C(10,0),C(8,8*C(11,0)

C(1,0)*C(4,1),C(2,1)*C(5,1),C(3,2)*C(6,1),C(4,3)*C(7,1), C(5,4)*C(8,1), C(6,5)*C(9,1),C(7,6)*C(10,1),C(8,7)*C(11,1)

C(2,0)*C(5,2),C(3,1)*C(6,2),C(4,2)*C(7,2),C(5,3)*C(8,2), C(6,4)*C(9,2),C(7,2)*C(10,2),C(8,6)*C(11,2)

C(3,0)*C(6,3),C(4,1)*C(7,3),C(5,2)*C(8,3),C(6,3)* C(9,3), C(7,4)*C(10,3),C(8,3)*C(11,3)

C(4,0)*C(7,4),C(5,1)*C(8,4),C(6,2)*C(9,4),C(7,3)*C(10,4), C(8,4)*C(11,4)

C(5,0)*C(8,5),C(6,1)*C(9,5),C(7,2)*C(10,5),C(8,3)*C(11,5)

C(6,0)*C(9,6),C(7,1)*C(10,6),C(8,2)*C(11,6)

C(7,0)*C(10,7),C(8,1)*C(11,7)

C(8,0)*C(11,8). (End)

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 *)

CROSSREFS

Cf. A000292.

Sequence in context: A277583 A213765 A182971 * A178216 A307529 A019213

Adjacent sequences:  A062142 A062143 A062144 * A062146 A062147 A062148

KEYWORD

nonn,tabl

AUTHOR

Wolfdieter Lang, Jun 19 2001

STATUS

approved

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Last modified January 21 11:09 EST 2020. Contains 331105 sequences. (Running on oeis4.)