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A062145
Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.
22
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
OFFSET
0,3
COMMENTS
Coefficient triangle of certain polynomials N(3; m,x).
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).
T(2*n, n) = (1/2)*(n+1)^2*A000108(n)*A000108(n+2).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^floor((n+2)/2)*(A047074(n+3) - A047074(n+ 2)). (End)
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).
Sequence in context: A213765 A349809 A182971 * A178216 A307529 A019213
KEYWORD
nonn,tabl,changed
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