A344059 Irregular triangle read by rows: T(n,k) = A344031(n,k)/2, n >= 1, 0 <= k <= 2*n-2.

1, 2, 0, -3, 4, -6, -18, 0, 18, 2, -48, -72, 126, 198, 0, -162, -44, -240, 210, 1620, 1260, -2484, -2700, 0, 1944, -328, -60, 5940, 11880, -13050, -45468, -24300, 52812, 44388, 0, -29160, -716, 11760, 53424, -27360, -360360, -450576, 555984, 1279152, 526176, -1242216, -857304, 0, 524880

Offset

1,2

Comments

The main entry is A344031, but someone may search for this sequence.

T(n,k) = [x^k]q_n(x), where 2 * (q_n(x)/x^(3n)) * exp(-1/x^2) is the n-th derivative of exp(-1/x^2).

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]q_n(x), where q_1(x) = 1, q_{n+1}(x) = (q_n(x))' - (3*n*x^2 - 2)*(q_n(x)), n >= 1.

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) = (-3)*(-6)*...*(-(3*n-3)) = (-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) = 2 * (-3)^(n-2) * A000254(n-1).

Example

q_1(x) = 1;
q_2(x) = -3*x^2 + 2;
q_3(x) = 18*x^4 - 18*x^2 - 6*x + 4;
q_4(x) = -162*x^6 + 198*x^4 + 126*x^3 - 72*x^2 - 48*x + 2;
q_5(x) = 1944*x^8 - 2700*x^6 - 2484*x^5 + 1260*x^4 + 1620*x^3 + 210*x^2 - 240*x - 44;
...

Prog

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

Crossrefs

Cf. A344031, A000254.

Sequence in context: A363346 A243202 A257136 * A258871 A283528 A336972

Adjacent sequences: A344056 A344057 A344058 * A344060 A344061 A344062

Keyword

sign,tabf

Author

Jianing Song, May 08 2021

Status

approved