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
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(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
T(n, k) = binomial(n, k)*binomial(n+5, k). - G. C. Greubel, Feb 28 2025
EXAMPLE
Triangle begins as:
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, 3003;
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 *)
A062190[n_, k_]:= Binomial[n, k]*Binomial[n+5, k];
Table[A062190[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 28 2025 *)
PROG
(Magma)
A062190:= func< n, k | Binomial(n, k)*Binomial(n+5, k) >;
[A062190(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2025
(SageMath)
def A062190(n, k): return binomial(n, k)*binomial(n+5, k)
print(flatten([[A062190(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 28 2025
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).
Columns k: A028557 (k=1), A104676 (k=2), A104677 (k=3), A104678 (k=4), A104679 (k=5), A104680 (k=6).
KEYWORD
nonn,tabl
AUTHOR
Wolfdieter Lang, Jun 19 2001
STATUS
approved