A194938 Triangle read by rows: coefficients of polynomials p(x,n) defined by 1/(1-t-t^2)^x = Sum_{n=1..oo} p(x,n)*t^n/n!.

1, 0, 1, 0, 3, 1, 0, 8, 9, 1, 0, 42, 59, 18, 1, 0, 264, 450, 215, 30, 1, 0, 2160, 4114, 2475, 565, 45, 1, 0, 20880, 43512, 30814, 9345, 1225, 63, 1, 0, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1, 0, 3064320, 7235568, 6316316, 2673972, 594489, 69552

Offset: 1

Author

Roger L. Bagula, Apr 17 2008

Keywords

• nonn
• tabl

Comments

A039692 is a similar triangle but without the leading column.

1/(1-t-t^2) is the g.f. for the Fibonacci numbers (A000045).

Row sums: A005442(n-1).

Also the Bell transform of n!*(F(n)+F(n+2)), F(n) the Fibonacci numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016

Examples

			Triangle begins
1;
0, 1;
0, 3, 1;
0, 8, 9, 1;
0, 42, 59, 18, 1;
0, 264, 450, 215, 30, 1;
0, 2160, 4114, 2475, 565, 45, 1;
0, 20880, 43512, 30814, 9345, 1225, 63, 1;
0, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1;
0, 3064320, 7235568, 6316316, 2673972, 594489, 69552, 4074, 108, 1;
0, 44634240, 110499696, 103889700, 49087520, 12803175, 1887753, 154350,6630,135,1;
		

References

• Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150

Crossrefs

Cf. A000045, A005442, A039692.

Programs

Maple

# The function BellMatrix is defined in A264428.
with(combinat): g := n -> factorial(n)*(fibonacci(n)+fibonacci(n+2)):
BellMatrix(g, 10); # Peter Luschny, Jan 21 2016

Mathematica

p[t_] = 1/(1 - t - t^2)^x; Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
(* Second program *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, n!*(Fibonacci[n] + Fibonacci[n+2])], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Sage

# uses[bell_matrix from A264428]
bell_matrix(lambda n: factorial(n)*(fibonacci(n)+fibonacci(n+2)), 8) # Peter Luschny, Jan 21 2016

Extensions

Edited by N. J. A. Sloane, Aug 28 2011