A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2*n} T(n,k) * x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.

1, 0, 0, 1, 0, -2, 0, 0, 1, 2, 0, 0, -6, 0, 0, 1, 0, 0, 20, 0, 0, -12, 0, 0, 1, 0, -40, 0, 0, 80, 0, 0, -20, 0, 0, 1, 40, 0, 0, -360, 0, 0, 220, 0, 0, -30, 0, 0, 1, 0, 0, 1120, 0, 0, -1680, 0, 0, 490, 0, 0, -42, 0, 0, 1, 0, -2240, 0, 0, 9520, 0, 0, -5600, 0, 0, 952, 0, 0, -56, 0, 0, 1

Offset: 0

Author

Werner Schulte, Jan 27 2020

Keywords

• sign
• tabf

Comments

Let r(s;n,x) = Sum_{k=0..s*n} A(s;n,k)*x^k = (-1)^n * e^(x^(s+1)/(s+1)) * (((d/dx)^n) e^(-x^(s+1)/(s+1))) for n >= 0 and x complex and some fixed integer s >= 1. Special cases: A(1;n,k) = A066325(n,k) and A(2;n,k) is this triangle. Formula: A(s;n,k) = (Sum_{i=0..floor(k/(s+1))} (-1)^i * binomial((n+k) /(s+1),i) * binomial(n+k-(s+1)*i,n)) * (-1)^(n-(n+k)/(s+1)) * (n!) / ((s+1)^((n+k)/(s+1)) * (((n+k)/(s+1))!)) if (n+k) mod (s+1) = 0 else 0 with n >= 0 and 0 <= k <= s*n.

Recurrence: (1) A(s;n,k) = A(s;n-1,k-s) - (k+1) * A(s;n-1,k+1),

(2) r(s;n,x) = x^s * r(s;n-1,x) - ((d/dx) r(s;n-1,x)) for n > 0 with initial values A(s;0,0) = 1 = r(s;0,x) and A(s;n,k) = 0 if k < 0 or k > s*n or (n+k) mod (s+1) > 0;

E.g.f.: Sum_{n>=0} r(s;n,x)*t^n/(n!) = e^((x^(s+1)-(x-t)^(s+1))/(s+1)).

This generalization is result of a long and intensive discussion with Wolfdieter Lang. For more information see A091752.

Examples

			The irregular triangle T(n,k) starts:
n\k:  0     1    2    3    4     5   6     7   8   9  10   . . .      16
========================================================================
0  :  1
1  :  0     0    1
2  :  0    -2    0    0    1
3  :  2     0    0   -6    0     0   1
4  :  0     0   20    0    0   -12   0     0   1
5  :  0   -40    0    0   80     0   0   -20   0   0   1
6  : 40     0    0 -360    0     0 220     0   0 -30   0   0 1
7  :  0     0 1120    0    0 -1680   0     0 490   0   0 -42 0   0 1
8  :  0 -2240    0    0 9520     0   0 -5600   0   0 952   0 0 -56 0 0 1
etc.
		

Crossrefs

Cf. A049404, A066325, A091752.

Row sums are (-1)^n*A252284(n).

Formula

T(n,k) = (-1)^k * (n!) * (Sum_{i=0..floor(k/3)} (-1)^i * binomial((n+k) /3,i) * binomial(n+k-3*i,n)) / (3^((n+k)/3) * ((n+k)/3)!) if (n+k) mod 3 = 0 else 0 with n >= 0 and 0 <= k <= 2*n.

Recurrence: (1) T(n,k) = T(n-1,k-2) - (k+1) * T(n-1,k+1),

(2) T(n,k) = T(n-1,k-2) - 2*(n-1)*T(n-2,k-1) + (n-1)*(n-2)*T(n-3,k),

(3) k*T(n,k) = 2*n*T(n-1,k-2) - n*(n-1)*T(n-2,k-1),

(4) p(n,x) = x^2 * p(n-1,x) - (d/dx) p(n-1,x),

(5) p(n,x) = x^2*p(n-1,x) - 2*(n-1)*x*p(n-2,x) + (n-1)*(n-2)*p(n-3,x),

(6) (d/dx) p(n,x) = 2*n*x*p(n-1,x) - n*(n-1)*p(n-2,x) for n > 0 with initial values T(0,0) = 1 = p(0,x) and T(n,k) = 0 if k < 0 or k > 2*n or (n+k) mod 3 > 0.

T(n,2*n) = 1 for n >= 0.

T(3*n,0) = -T(3*n-1,1) = 2*T(3*n-2,2) = ((3*n)!)/(3^n * (n!)) for n > 0.

The polynomials p(n,x) satisfy for n >= 0 and x complex the differential equation: 0 = (((d/dx)^3) p(n,x)) - 2*x^2*(((d/dx)^2) p(n,x)) + (x^4 + 2*(n-1)*x) * ((d/dx) p(n,x)) - (2*n*x^3-(n+3)*n) * p(n,x).

E.g.f.: Sum_{n>=0} p(n,x)*t^n/(n!) = e^((x^3-(x-t)^3)/3).

((d/dx)^m) p(n,x) = Sum_{i=0..m} (-1)^i * binomial(m,i) * p(m-i,-x) * p(n+i,x) for m,n >= 0 and x complex.

T(3*n-k,k) = A091752(n+1,k+2) for 0 <= k <= 2*n.

(-1)^(n-k) * T(n,3*k-n) = A049404(n,k) for n > 0 and (n+2)/3 <= k <= n.