A344059 - OEIS

%I #18 Jan 05 2024 14:57:36

%S 1,2,0,-3,4,-6,-18,0,18,2,-48,-72,126,198,0,-162,-44,-240,210,1620,

%T 1260,-2484,-2700,0,1944,-328,-60,5940,11880,-13050,-45468,-24300,

%U 52812,44388,0,-29160,-716,11760,53424,-27360,-360360,-450576,555984,1279152,526176,-1242216,-857304,0,524880

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

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

%C 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).

%H Jianing Song, <a href="/A344059/b344059.txt">Rows n = 1..100, flattened</a> (T(n,k) occurs at the position (n-1)^2 + k + 1.)

%F 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.

%F 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.

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

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

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

%e q_1(x) = 1;

%e q_2(x) = -3*x^2 + 2;

%e q_3(x) = 18*x^4 - 18*x^2 - 6*x + 4;

%e q_4(x) = -162*x^6 + 198*x^4 + 126*x^3 - 72*x^2 - 48*x + 2;

%e q_5(x) = 1944*x^8 - 2700*x^6 - 2484*x^5 + 1260*x^4 + 1620*x^3 + 210*x^2 - 240*x - 44;

%e ...

%o (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

%o T(n, k) = polcoeff(up_to_N_reduced(n)[n], k)

%Y Cf. A344031, A000254.

%K sign,tabf

%O 1,2

%A _Jianing Song_, May 08 2021