A062196 - OEIS

%I #37 Feb 13 2024 07:35:17

%S 1,1,3,1,8,6,1,15,30,10,1,24,90,80,15,1,35,210,350,175,21,1,48,420,

%T 1120,1050,336,28,1,63,756,2940,4410,2646,588,36,1,80,1260,6720,14700,

%U 14112,5880,960,45,1,99,1980,13860,41580,58212,38808,11880,1485,55

%N Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

%C 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).

%F T(m, k) = [x^k]N(2; m, x), with N(2; m, x) = ((1-x)^(3+2*m))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+3))).

%F 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).

%F T(n,m) = binomial(n, m)*(binomial(n+1, m) + binomial(n+1, m-1)). - _Vladimir Kruchinin_, Apr 06 2018

%e Triangle starts:

%e [0] 1;

%e [1] 1, 3;

%e [2] 1, 8,  6;

%e [3] 1, 15, 30,   10;

%e [4] 1, 24, 90,   80,    15;

%e [5] 1, 35, 210,  350,   175,   21;

%e [6] 1, 48, 420,  1120,  1050,  336,   28;

%e [7] 1, 63, 756,  2940,  4410,  2646,  588,   36;

%e [8] 1, 80, 1260, 6720,  14700, 14112, 5880,  960,   45;

%e [9] 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55.

%p T := (n, k) -> binomial(n, k)*binomial(n + 2, k);

%p seq(seq(T(n, k), k=0..n), n=0..9); # _Peter Luschny_, Sep 30 2021

%Y Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), this sequence (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).

%Y Cf. A089849, A001791.

%K nonn,easy,tabl

%O 0,3

%A _Wolfdieter Lang_, Jun 19 2001

%E New name by _Peter Luschny_, Sep 30 2021