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A062190 Coefficient triangle of certain polynomials N(5; m,x). 26
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 (list; table; graph; refs; listen; history; text; internal format)
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.

From Zerinvary Lajos, Apr 01 2005: (Start)

Formatted as a square array:

C(0,0)*C(5,0), C(1,1)*C(6,0), C(2,2)*C(7,0), C(3,3)*C(8,0), C(4,4)*C(9,0), C(5,5)*C(10,0), C(6,6)*C(11,0), C(7,7)*C(12,0), C(8,8)*C(13,0)

C(1,0)*C(6,1), C(2,1)*C(7,1), C(3,2)*C(8,1), C(4,3)*C(9,1), C(5,4)*C(10,1), C(6,5)*C(11,1), C(7,6)*C(12,1), C(8,7)*C(13,1)

C(2,0)*C(7,2), C(3,1)*C(8,2), C(4,2)*C(9,2), C(5,3)*C(10,2), C(6,4)*C(11,2), C(7,5)*C(12,2), C(8,6)*C(13,2)

C(3,0)*C(8,3), C(4,1)*C(9,3), C(5,2)*C(10,3), C(6,3)*C(11,3), C(7,4)*C(12,3), C(8,5)*C(13,3)

C(4,0)*C(9,4), C(5,1)*C(10,4), C(6,2)*C(11,4), C(7,3)*C(12,4), C(8,4)*C(13,4)

C(5,0)*C(10,5), C(6,1)*C(11,5), C(7,2)*C(12,5), C(8,3)*C(13,5)

C(6,0)*C(11,6), C(7,1)*C(12,6), C(8,2)*C(13,6)

C(7,0)*C(12,7), C(8,1)*C(13,7)

C(8,0)*C(13,8).

(End)

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

Cf. A062145, 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

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Last modified February 27 17:59 EST 2020. Contains 332307 sequences. (Running on oeis4.)