A391950 Triangle T(n,k) of coefficients of polynomials arising from Gaussian-weighted Chebyshev integrals.

2, 12, 0, 120, 8, 2, 1680, 160, 48, 0, 30240, 3360, 1064, 24, 2, 665280, 80640, 26208, 960, 108, 0, 17297280, 2217600, 731808, 33696, 4240, 48, 2, 518918400, 69189120, 23063040, 1216512, 162624, 3200, 192, 0, 17643225600, 2421619200, 812972160, 46949760, 6520800, 168672, 11760, 80, 2

Offset

1,1

Comments

Triangular sequence, read by rows, of coefficients of polynomials P_n(x) = Sum_{k = 0..n-1} T(n,k)*x^(2*k+1), defined by the following recursive relation: P_1(x) = 2*x; P_n(x) = 2*x^(2*n-1) - x^2*P_{n-1}(x) + 2*(2*n-1) * Sum_{j = 1..n-1} x^(2*n-2*j-2)*P_j(x).

These polynomials appear in the explicit expression of Gaussian-weighted integrals of Chebyshev polynomials of the first kind T_n: L^(2*n+1) * Integral_{x=-1..1} e^-(L*x)^2 * T_{2*n}(x) dx = -P_n(L)*e^(-L^2) + Q_n(L)*sqrt(Pi)*erf(L) for L>0 and n>1.

Polynomials Q_n seem to satisfy Q_n(x) = Sum_{k = 0..n} (-1)^k*A370706(n*(n+1)/2+n-k)*x^(2*k).

Links

Table of n, a(n) for n=1..45.

Formula

T(n,k) = 2*delta(n-1,k) - T(n-1,k-1) + 2*(2*n-1) * Sum_{j = 1..n-1} T(j,k-(n-j-1)), for n >= 1 and 0 <= k < n and 0 elsewhere.

Example

The first rows of the triangle are:
          x     x^3    x^5  x^7 x^9
P_1:      2;
P_2:     12;     0;
P_3:    120;     8;     2;
P_4:   1680;   160;    48;   0;
P_5:  30240;  3360;  1064;  24;  2;
...

Mathematica

c[n_Integer?Positive, k_Integer] /; 0 <= k < n := c[n, k] = Module[{c1, c2, c3}, c1 = If[k == n - 1, 2, 0]; c2 = If[k >= 1, -c[n - 1, k - 1], 0]; c3 = 2 (2 n - 1) Sum[If[k >= n - j - 1, c[j, k - (n - j - 1)], 0], {j, 1, n - 1}]; c1 + c2 + c3]; c[_, _] := 0; Flatten[Table[c[n, k], {n, 1, 9}, {k, 0, n - 1}]]

Crossrefs

Cf. A370706.

Sequence in context: A327780 A350258 A364781 * A166489 A160367 A016736

Adjacent sequences: A391947 A391948 A391949 * A391951 A391952 A391953

Keyword

nonn,tabl

Author

Gabriele Celebre, Dec 23 2025

Status

approved