A344031 Irregular triangle read by rows: T(n,k) = [x^k]p_n(x), where (p_n(x)/x^(3n)) * exp(-1/x^2) is the n-th derivative of exp(-1/x^2), n >= 1, 0 <= k <= 2*n-2.

2, 4, 0, -6, 8, -12, -36, 0, 36, 4, -96, -144, 252, 396, 0, -324, -88, -480, 420, 3240, 2520, -4968, -5400, 0, 3888, -656, -120, 11880, 23760, -26100, -90936, -48600, 105624, 88776, 0, -58320, -1432, 23520, 106848, -54720, -720720, -901152, 1111968, 2558304, 1052352, -2484432, -1714608, 0, 1049760

Offset

1,1

Comments

p_n(x) is a polynomial of degree 2*n - 2 for n >= 1.

Note that the n-th derivative of f(x) = exp(-1/x^2), x != 0; 0, x = 0 is f^(n)(x) = (p_n(x)/x^(3n)) * exp(-1/x^2), x != 0; 0, x = 0. f(x) is an example of smooth function which is not analytic at x = 0.

Links

Jianing Song, Rows n = 1..100, flattened (T(n,k) occurs at the position (n-1)^2 + k + 1.)

Formula

T(n,k) = [x^k]p_n(x), where p_0(x) = 1, p_{n+1}(x) = (p_n(x))' - (3*n*x^2 - 2)*(p_n(x)), n >= 0.

For n >= 1, T(n+1,k) = (k+1)*T(n,k+1) + 2*T(n,k) - 3*n*T(n,k-2) for 0 <= k <= 2*n, where T(n,-2) = T(n,-1) = T(n,2*n-1) = T(n,2*n) = T(n,2*n+1) = 0.

For n >= 1, T(n,2*n-2) = 2*(-3)*(-6)*...*(-(3*n-3)) = 2 * (-3)^(n-1) * (n-1)!.

For n >= 2, T(n,2*n-3) = 0. It seems that T(n,k) != 0 for k != 2*n-3.

For n >= 2, T(n,2*n-4) = 4 * (-3)^(n-2) * A000254(n-1).

Example

p_0(x) = 1;
p_1(x) = 2;
p_2(x) = -6*x^2 + 4;
p_3(x) = 36*x^4 - 36*x^2 - 12*x + 8;
p_4(x) = -324*x^6 + 396*x^4 + 252*x^3 - 144*x^2 - 96*x + 4;
p_5(x) = 3888*x^8 - 5400*x^6 - 4968*x^5 + 2520*x^4 + 3240*x^3 + 420*x^2 - 480*x - 88;
...

Prog

(PARI) up_to_N(n) = my(v=vector(n+1)); v[1]=1; for(k=1, n, v[k+1] = v[k]' - (3*(k-1)*x^2-2)*v[k]); v
T(n, k) = polcoeff(up_to_N(n)[n+1], k)

Crossrefs

Cf. A000254.

Sequence in context: A131715 A200165 A326938 * A229534 A021810 A355007

Adjacent sequences: A344028 A344029 A344030 * A344032 A344033 A344034

Keyword

sign,tabf

Author

Jianing Song, May 07 2021

Status

approved