%I #25 Feb 13 2024 07:53:52
%S 1,1,6,1,14,21,1,24,84,56,1,36,216,336,126,1,50,450,1200,1050,252,1,
%T 66,825,3300,4950,2772,462,1,84,1386,7700,17325,16632,6468,792,1,104,
%U 2184,16016,50050,72072,48048,13728
%N Coefficient triangle of certain polynomials N(5; m,x).
%C 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.
%F 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))).
%F 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).
%F 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]
%e 1,
%e 1, 6,
%e 1, 14, 21,
%e 1, 24, 84, 56,
%e 1, 36, 216, 336, 126,
%e 1, 50, 450, 1200, 1050, 252,
%e 1, 66, 825, 3300, 4950, 2772, 462,
%e 1, 84, 1386, 7700, 17325, 16632, 6468, 792,
%e 1, 104, 2184, 16016, 50050, 72072, 48048, 13728, 1287,
%e 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002,
%e 1, 150, 4725, 54600, 286650, 756756, 1051050, 772200, 289575, 50050, ...
%p A062190 := proc(m,k)
%p add( (binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j,j=0..m) ;
%p coeftayl(%,x=0,k) ;
%p end proc: # _R. J. Mathar_, Nov 29 2015
%t 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 *)
%Y Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), this sequence (c=6).
%Y Cf. A028557 (k=1), A104676 (k=2), A104677 (k=3).
%K nonn,tabl
%O 0,3
%A _Wolfdieter Lang_, Jun 19 2001