OFFSET
0,3
COMMENTS
Coefficient triangle of certain polynomials N(3; m,x).
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=3) Laguerre triangle L(3; n+m, m) = A062137(n+m, m), n >= 0, is N(3; m, x)/(1-x)^(2*(m+2)), with the row polynomials N(3; m, x) := Sum_{k=0..m} a(m, k)*x^k.
N(3; m, x) := ((1-x)^(2*(m+2)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+4))); a(m, k) = [x^k]N(3; m, x).
N(3; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+3-j)!/((m+3)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(3; m, x)= x^m*(2*m+3)! * 2F1(-m, -m; -2*m-3; (x-1)/x)/((m+3)!*m!). - Jean-François Alcover, Sep 18 2013
From G. C. Greubel, Mar 07 2025 : (Start)
T(n, k) = binomial(n, k)*binomial(n+3, k).
EXAMPLE
As a square array:
1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
4, 10, 18, 28, 40, 54, 70, 88, ... A028552;
10, 45, 126, 280, 540, 945, 1540, ....... A105938;
20, 140, 560, 1680, 4200, 9240, ............. A105939;
35, 350, 1890, 7350, 23100, 62370, ............. A027803;
56, 756, 5292, 25872, 99792, .................... A105940;
84, 1470, 12936, 77616, ........................... A105942;
120, 2640, 28512, .................................. A105943;
165, 4455, 57015, .................................. A105944;
....;
As a triangle:
1;
1, 4;
1, 10, 10;
1, 18, 45, 20;
1, 28, 126, 140, 35;
1, 40, 280, 560, 350, 56;
1, 54, 540, 1680, 1890, 756, 84;
1, 70, 945, 4200, 7350, 5292, 1470, 120;
1, 88, 1540, 9240, 23100, 25872, 12936, 2640, 165;
1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220;
....;
MATHEMATICA
NN[3, m_, x_] := x^m*(2*m+3)!*Hypergeometric2F1[-m, -m, -2*m-3, (x-1)/x]/( (m+3)!*m!); Table[CoefficientList[NN[3, m, x], x], {m, 0, 9}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
P[c_, n_, z_] := Sum[Binomial[n, k] Pochhammer[n-k+c, k] z^k /k!, {k, 0, n}];
CL[c_] := Table[CoefficientList[P[c, n, z], z], {n, 0, 5}] // TableForm
CL[4] (* Peter Luschny, Feb 12 2024 *)
A062145[n_, k_]:= Binomial[n, k]*Binomial[n+3, k];
Table[A062145[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 07 2025 *)
PROG
(Magma)
A062145:= func< n, k | Binomial(n, k)*Binomial(n+3, k) >;
[A062145(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 07 2025
(SageMath)
def A062145(n, k): return binomial(n, k)*binomial(n+3, k)
print(flatten([[A062145(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 07 2025
CROSSREFS
Family of polynomials: A008459 (c=1), A132813 (c=2), A062196 (c=3), this sequence (c=4), A062264 (c=5), A062190 (c=6).
Columns: A028552 (k=1), A105938 (k=2), A105939 (k=3), A027803 (k=4), A105940 (k=5), A105942 (k=6), A105943 (k=7), A105944 (k=8).
Diagonals: A000292 (k=n), A027800 (k=n-1), A107417 (k=n-2), A107418 (k=n-3), A107419 (k=n-4), A107420 (k=n-5), A107421 (k=n-6), A107422 (k=n-7).
Sums: A002054 (row).
KEYWORD
AUTHOR
Wolfdieter Lang, Jun 19 2001
EXTENSIONS
New name by Peter Luschny, Feb 12 2024
More terms from G. C. Greubel, Mar 07 2025
STATUS
approved