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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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
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.
Sequence in context: A123360 A072015 A123242 * A217205 A358566 A139769
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 06 2019
STATUS
approved

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Last modified April 21 23:54 EDT 2024. Contains 371886 sequences. (Running on oeis4.)