A322456 Triangle of Touchard's chord enumerating polynomial coefficients [x^k] P_n(x).

1, 1, 1, 3, 1, 4, 10, 12, 1, 5, 15, 35, 60, 77, 55, 1, 6, 21, 56, 126, 240, 391, 546, 624, 546, 273, 1, 7, 28, 84, 210, 462, 910, 1619, 2618, 3857, 5138, 6125, 6405, 5600, 3740, 1428, 1, 8, 36, 120, 330, 792, 1716, 3416, 6308, 10872, 17564, 26664, 38030, 50864, 63580, 73848, 78880, 76296, 65076, 46512, 25194, 7752

Offset

1,4

Links

Table of n, a(n) for n=1..64.

J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25, P_n(x).

Formula

T(n,k) = [x^k] P_n(x), k=n*(n-1)/2 down to k=n-1.

Example

The triangle starts
1;
1;
1,  3;
1,  4,   10,   12;
1,  5,   15,   35,   60,   77,   55;
1,  6,   21,   56,  126,  240,  391,  546,  624,  546,  273;
1,  7,   28,   84,  210,  462,  910, 1619, 2618, 3857, 5138, 6125, 6405, 5600, 3740, 1428;

Crossrefs

Cf. A322398 (S_n(x)), A000699 (row sums), A232223 (transpose)

Sequence in context: A178300 A081720 A137405 * A301701 A262078 A121922

Adjacent sequences: A322453 A322454 A322455 * A322457 A322458 A322459

Keyword

nonn,tabf

Author

R. J. Mathar, Dec 09 2018

Status

approved