This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A322941 #7 Jan 06 2019 11:24:55 %S A322941 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 A322941 10,1,208,628,796,568,252,71,12,1,568,1880,2680,2208,1170,414,97,14,1, %U A322941 1552,5584,8832,8232,5052,2140,632,127,16,1,4240,16480,28608,29712,20676,10160,3598,914,161,18,1 %N 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. %F A322941 p(n, x) = (x+2)*p(n-1, x) + 2*p(n-2, x) for n >= 3. %F A322941 T(n, k) = [x^k] p(n, x). %e A322941 The first few polynomials are: %e A322941 [0] p(0, x) = 1; %e A322941 [1] p(1, x) = x + 1; %e A322941 [2] p(2, x) = x^2 + 2*x + 1; %e A322941 [3] p(3, x) = x^3 + 4*x^2 + 7*x + 4; %e A322941 [4] p(4, x) = x^4 + 6*x^3 + 17*x^2 + 22*x + 10; %e A322941 [5] p(5, x) = x^5 + 8*x^4 + 31*x^3 + 64*x^2 + 68*x + 28; %e A322941 [6] p(6, x) = x^6 + 10*x^5 + 49*x^4 + 138*x^3 + 230*x^2 + 208*x + 76; %e A322941 The triangle starts: %e A322941 [0] 1; %e A322941 [1] 1, 1; %e A322941 [2] 1, 2, 1; %e A322941 [3] 4, 7, 4, 1; %e A322941 [4] 10, 22, 17, 6, 1; %e A322941 [5] 28, 68, 64, 31, 8, 1; %e A322941 [6] 76, 208, 230, 138, 49, 10, 1; %e A322941 [7] 208, 628, 796, 568, 252, 71, 12, 1; %e A322941 [8] 568, 1880, 2680, 2208, 1170, 414, 97, 14, 1; %e A322941 [9] 1552, 5584, 8832, 8232, 5052, 2140, 632, 127, 16, 1; %p A322941 p := proc(n) option remember; %p A322941 `if`(n < 3, [1, x+1, x^2 + 2*x + 1][n+1], (x+2)*p(n-1) + 2*p(n-2)); %p A322941 sort(expand(%)) end: seq(print(p(n)), n=0..11); # Computes the polynomials. %p A322941 seq(seq(coeff(p(n), x, k), k=0..n), n=0..10); %Y A322941 Row sums are A322940, alternating row sums are A000007. %Y A322941 Cf. A026150, A322942. %K A322941 nonn,tabl %O A322941 0,5 %A A322941 _Peter Luschny_, Jan 06 2019