A322939
a(n) = [x^n] (4*x^2 + x - 1)/(4*x^3 + 3*x^2 + 2*x - 1).
Original entry on oeis.org
1, 1, 1, 9, 25, 81, 273, 889, 2921, 9601, 31521, 103529, 340025, 1116721, 3667633, 12045529, 39560841, 129928801, 426722241, 1401474249, 4602830425, 15116972561, 49648333393, 163058906169, 535530702761, 1758831457601, 5776490648161, 18971598480169, 62307994735225
Offset: 0
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gf := (4*x^2 + x - 1)/(4*x^3 + 3*x^2 + 2*x - 1): ser := series(gf, x, 22):
seq(coeff(ser, x, n), n = 0..20);
a := proc(n) option remember;
`if`(n < 3, [1, 1, 1][n+1], 4*a(n-3) + 3*a(n-2) + 2*a(n-1)) end:
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.
Original entry on oeis.org
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
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;
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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);
Showing 1-2 of 2 results.