A322941 Coefficients of orthogonal polynomials p(n, x) where p(n, 0) is A026150 with 1 prepended. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 1, 1, 2, 1, 4, 7, 4, 1, 10, 22, 17, 6, 1, 28, 68, 64, 31, 8, 1, 76, 208, 230, 138, 49, 10, 1, 208, 628, 796, 568, 252, 71, 12, 1, 568, 1880, 2680, 2208, 1170, 414, 97, 14, 1, 1552, 5584, 8832, 8232, 5052, 2140, 632, 127, 16, 1, 4240, 16480, 28608, 29712, 20676, 10160, 3598, 914, 161, 18, 1

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

p(n, x) = (x+2)*p(n-1, x) + 2*p(n-2, x) for n >= 3.

T(n, k) = [x^k] p(n, x).

Example

The first few polynomials are:
[0] p(0, x) = 1;
[1] p(1, x) = x + 1;
[2] p(2, x) = x^2 +  2*x + 1;
[3] p(3, x) = x^3 +  4*x^2 +  7*x + 4;
[4] p(4, x) = x^4 +  6*x^3 + 17*x^2 +  22*x + 10;
[5] p(5, x) = x^5 +  8*x^4 + 31*x^3 +  64*x^2 +  68*x + 28;
[6] p(6, x) = x^6 + 10*x^5 + 49*x^4 + 138*x^3 + 230*x^2 + 208*x + 76;
The triangle starts:
[0]    1;
[1]    1,    1;
[2]    1,    2,    1;
[3]    4,    7,    4,    1;
[4]   10,   22,   17,    6,    1;
[5]   28,   68,   64,   31,    8,    1;
[6]   76,  208,  230,  138,   49,   10,   1;
[7]  208,  628,  796,  568,  252,   71,  12,   1;
[8]  568, 1880, 2680, 2208, 1170,  414,  97,  14,  1;
[9] 1552, 5584, 8832, 8232, 5052, 2140, 632, 127, 16, 1;

Maple

p := proc(n) option remember;
`if`(n < 3, [1, x+1, x^2 + 2*x + 1][n+1], (x+2)*p(n-1) + 2*p(n-2));
sort(expand(%)) end: seq(print(p(n)), n=0..11); # Computes the polynomials.
seq(seq(coeff(p(n), x, k), k=0..n), n=0..10);

Crossrefs

Row sums are A322940, alternating row sums are A000007.

Cf. A026150, A322942.

Sequence in context: A123360 A072015 A123242 * A217205 A358566 A139769

Adjacent sequences: A322938 A322939 A322940 * A322942 A322943 A322944

Keyword

nonn,tabl

Author

Peter Luschny, Jan 06 2019

Status

approved