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 A062145 Coefficient triangle of certain polynomials N(3; m,x). 19

%I

%S 1,1,4,1,10,10,1,18,45,20,1,28,126,140,35,1,40,280,560,350,56,1,54,

%T 540,1680,1890,756,84,1,70,945,4200,7350,5292,1470,120,1,88,1540,9240,

%U 23100,25872,12936,2640,165,1,108

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

%C From _Zerinvary Lajos_, Mar 24 2005: (Start)

%C Formatted as an upper right triangle:

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

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

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

%F 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.

%F 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).

%F 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).

%F 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]

%e From _Zerinvary Lajos_, Jan 02 2006: (Start)

%e As a square array:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 4, 10, 18, 28, 40, 54, 70, 88, ...

%e 10, 45, 126, 280, 540, 945, 1540, ...

%e 20, 140, 560, 1680, 4200, 9240, ...

%e 35, 350, 1890, 7350, 23100, ...

%e 56, 756, 5292, 25872, ...

%e ... (End)

%t 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 *)

%Y Cf. A000292.

%K nonn,tabl

%O 0,3

%A _Wolfdieter Lang_, Jun 19 2001

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Last modified February 29 08:26 EST 2020. Contains 332355 sequences. (Running on oeis4.)