OFFSET
0,3
COMMENTS
Also the coefficient triangle of certain polynomials N(2; m,x) := Sum_{k=0..m} T(m,k)*x^k. The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=2) Laguerre triangle L(2; n+m,m) = A062139(n+m,m), n >= 0, is N(2; m,x)/(1-x)^(3+2*m), with the row polynomials N(2; m,x).
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(m, k) = [x^k] N(2; m, x), where N(2; m, x) = ((1-x)^(3+2*m))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+3))).
N(2; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+2-j)!/((m+2)!*(m-j)!)*(x^(m-j)))*(1-x)^j).
T(n,m) = binomial(n, m)*(binomial(n+1, m) + binomial(n+1, m-1)). - Vladimir Kruchinin, Apr 06 2018
From G. C. Greubel, Feb 21 2025: (Start)
EXAMPLE
Triangle starts:
n\k 0...1.....2......3..... 4.....;
[0] 1;
[1] 1, 3;
[2] 1, 8, 6;
[3] 1, 15, 30, 10;
[4] 1, 24, 90, 80, 15;
[5] 1, 35, 210, 350, 175, 21;
[6] 1, 48, 420, 1120, 1050, 336, 28;
[7] 1, 63, 756, 2940, 4410, 2646, 588, 36;
[8] 1, 80, 1260, 6720, 14700, 14112, 5880, 960, 45;
[9] 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55.
MAPLE
T := (n, k) -> binomial(n, k)*binomial(n + 2, k);
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Sep 30 2021
MATHEMATICA
A062196[n_, k_]:= Binomial[n, k]*Binomial[n+2, k];
Table[A062196[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 21 2025 *)
PROG
(Magma)
A062196:= func<n, k | Binomial(n, k)*Binomial(n+2, k) >;
[A062196(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 21 2025
(SageMath)
def A062196(n, k): return binomial(n, k)*binomial(n+2, k)
print(flatten([[A062196(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 21 2025
CROSSREFS
Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), this sequence (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Diagonals: A000217 (k=n), A002417 (k=n-1), A001297 (k=n-2), A105946 (k=n-3), A105947 (k=n-4), A105948 (k=n-5), A107319 (k=n-6).
KEYWORD
AUTHOR
Wolfdieter Lang, Jun 19 2001
EXTENSIONS
New name by Peter Luschny, Sep 30 2021
STATUS
approved