A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.

1, 1, 1, 1, 4, 2, 1, 10, 16, 6, 1, 20, 70, 76, 24, 1, 35, 224, 496, 428, 120, 1, 56, 588, 2260, 3808, 2808, 720, 1, 84, 1344, 8140, 23008, 32152, 21096, 5040, 1, 120, 2772, 24772, 107328, 245560, 298688, 178848, 40320

Offset: 0

Author

Michael Somos, Mar 15 2023

Keywords

• nonn
• tabl

Comments

This sequence is essentially A204024 except for extra row, alternating signs and reversed rows.

The sequence of polynomials a(m) satisfies a(m)*a(m-2) = a(m-1) * (a(m-1) + x*a(m-2) + a(m-3)) - a(m-2)^2 for all m > 3.

Examples

			a(3) = 1, a(4) = 1 + x, a(5) = 1 + 4*x + 2*x^2.
Triangular array T(n, k) starts:
n\k | 0   1   2   3   4   5
--- + - --- --- --- --- ---
 0  | 1
 1  | 1   1
 2  | 1   4   2
 3  | 1  10  16   6
 4  | 1  20  70  76  24
 5  | 1  35 224 496 428 120
		

Crossrefs

Cf. A000292, A040977, A058797, A093986, A204024.

Programs

Mathematica

T[ n_, k_] := If[ n<0, 0, Module[{a = Table[1, n+3], x}, Do[ a[[m]] = a[[m-1]] *(a[[m-1]] + x*a[[m-2]] + a[[m-3]])/a[[m-2]] - a[[m-2]] //Factor//Expand, {m, 4, n+3}]; Coefficient[ a[[n+3]], x, k]]];

PARI

{T(n, k) = if( n<0, 0, my(a = vector(n+3, i, 1)); for(m = 4, n+3, a[m] = a[m-1]*(a[m-1] + 'x*a[m-2] + a[m-3])/a[m-2] - a[m-2]); polcoeff( a[n+3], k))};

Formula

If x=1, then a(n) = A058797(n+2) = Sum_{k=0..n} T(n, k).

If x=2, then a(n) = A093986(n+2).

T(n, n) = n!, T(n, 0) = 1, T(n, 1) = A000292(n). T(n, 2) = 2*A040977(n-2).