OFFSET
0,3
COMMENTS
The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=5) Laguerre triangle L(5; n+m,m)= A062138(n+m,m), n >= 0, is N(5; m,x)/(1-x)^(2*(m+3)), with the row polynomials N(5; m,x) := Sum_{k=0..m} a(m,k)*x^k.
FORMULA
a(m, k) = [x^k]N(5; m, x), with N(5; m, x) = ((1-x)^(2*(m+3)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+6))).
N(5; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(5; m, x)= x^m*(2*m+5)! * 2F1(-m, -m; -2*m-5; (x-1)/x)/((m+5)!*m!). [Jean-François Alcover, Sep 18 2013]
EXAMPLE
1,
1, 6,
1, 14, 21,
1, 24, 84, 56,
1, 36, 216, 336, 126,
1, 50, 450, 1200, 1050, 252,
1, 66, 825, 3300, 4950, 2772, 462,
1, 84, 1386, 7700, 17325, 16632, 6468, 792,
1, 104, 2184, 16016, 50050, 72072, 48048, 13728, 1287,
1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002,
1, 150, 4725, 54600, 286650, 756756, 1051050, 772200, 289575, 50050, ...
MAPLE
A062190 := proc(m, k)
add( (binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j, j=0..m) ;
coeftayl(%, x=0, k) ;
end proc: # R. J. Mathar, Nov 29 2015
MATHEMATICA
NN[5, m_, x_] := x^m*(2*m+5)!*Hypergeometric2F1[-m, -m, -2*m-5, (x-1)/x]/((m+5)!*m!); Table[CoefficientList[NN[5, m, x], x], {m, 0, 8}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wolfdieter Lang, Jun 19 2001
STATUS
approved