A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

1, 1, 3, 1, 8, 6, 1, 15, 30, 10, 1, 24, 90, 80, 15, 1, 35, 210, 350, 175, 21, 1, 48, 420, 1120, 1050, 336, 28, 1, 63, 756, 2940, 4410, 2646, 588, 36, 1, 80, 1260, 6720, 14700, 14112, 5880, 960, 45, 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55

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

Table of n, a(n) for n=0..54.

Formula

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))).

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

Example

Triangle starts:
[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

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).

Cf. A089849, A001791.

Sequence in context: A258205 A258018 A188939 * A103247 A030523 A207815

Adjacent sequences: A062193 A062194 A062195 * A062197 A062198 A062199

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Jun 19 2001

Extensions

New name by Peter Luschny, Sep 30 2021

Status

approved