A062190 Coefficient triangle of certain polynomials N(5; m,x).

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

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.

Links

Table of n, a(n) for n=0..43.

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

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), this sequence (c=6).

Cf. A028557 (k=1), A104676 (k=2), A104677 (k=3).

Sequence in context: A122508 A171006 A176121 * A080211 A146997 A147483

Adjacent sequences: A062187 A062188 A062189 * A062191 A062192 A062193

Keyword

nonn,tabl

Author

Wolfdieter Lang, Jun 19 2001

Status

approved