A322941 - OEIS

%I #7 Jan 06 2019 11:24:55

%S 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,

%T 10,1,208,628,796,568,252,71,12,1,568,1880,2680,2208,1170,414,97,14,1,

%U 1552,5584,8832,8232,5052,2140,632,127,16,1,4240,16480,28608,29712,20676,10160,3598,914,161,18,1

%N 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.

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

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

%e The first few polynomials are:

%e [0] p(0, x) = 1;

%e [1] p(1, x) = x + 1;

%e [2] p(2, x) = x^2 +  2*x + 1;

%e [3] p(3, x) = x^3 +  4*x^2 +  7*x + 4;

%e [4] p(4, x) = x^4 +  6*x^3 + 17*x^2 +  22*x + 10;

%e [5] p(5, x) = x^5 +  8*x^4 + 31*x^3 +  64*x^2 +  68*x + 28;

%e [6] p(6, x) = x^6 + 10*x^5 + 49*x^4 + 138*x^3 + 230*x^2 + 208*x + 76;

%e The triangle starts:

%e [0]    1;

%e [1]    1,    1;

%e [2]    1,    2,    1;

%e [3]    4,    7,    4,    1;

%e [4]   10,   22,   17,    6,    1;

%e [5]   28,   68,   64,   31,    8,    1;

%e [6]   76,  208,  230,  138,   49,   10,   1;

%e [7]  208,  628,  796,  568,  252,   71,  12,   1;

%e [8]  568, 1880, 2680, 2208, 1170,  414,  97,  14,  1;

%e [9] 1552, 5584, 8832, 8232, 5052, 2140, 632, 127, 16, 1;

%p p := proc(n) option remember;

%p `if`(n < 3, [1, x+1, x^2 + 2*x + 1][n+1], (x+2)*p(n-1) + 2*p(n-2));

%p sort(expand(%)) end: seq(print(p(n)), n=0..11); # Computes the polynomials.

%p seq(seq(coeff(p(n), x, k), k=0..n), n=0..10);

%Y Row sums are A322940, alternating row sums are A000007.

%Y Cf. A026150, A322942.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Jan 06 2019