A208744 Triangle relating to ordered Bell numbers, A000670.

1, 1, 2, 1, 3, 9, 1, 4, 18, 52, 1, 5, 30, 130, 375, 1, 6, 45, 260, 1125, 3246, 1, 7, 63, 455, 2625, 11361, 32781, 1, 8, 84, 728, 5250, 30296, 131124, 378344, 1, 9, 108, 1092, 9450, 68166, 393372, 1702548, 4912515, 1, 10, 135, 1560, 15750, 136332, 983430, 5675160, 24562575, 70872610

Offset: 1

Author

Gary W. Adamson, Mar 05 2012

Keywords

• nonn
• tabl

Comments

Row sums = A000670 starting (1, 3, 13, 75,...).

Right border = A052882 starting (1, 2, 9, 52, 375,...).

A000670 is the eigensequence of triangle A074909, deleting the first "1".

Triangle A074909 is the "beheaded" Pascal's triangle: (1; 1,2; 1,3,3;...).

Examples

			Row 4 (nonzero terms) = (1, 4, 18, 52) = termwise product of (1, 4, 6, 4) and (1, 1, 3, 13).
First few rows of the triangle:
1;
1, 2;
1, 3, 9;
1, 4, 18, 52;
1, 5, 30, 130, 375;
1, 6, 45, 260, 1125, 3246;
1, 7, 63, 455, 2625, 11361, 32781;
1, 8, 84, 728, 5250, 30296, 131124, 378344;
...
		

Crossrefs

Cf. A000670, A052882, A154921.

Programs

Mathematica

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i + k + r)*(i + r)^(n - r)/(i!*(k - i - r)!), {i, 0, k - r}], {k, r, n}]; Fubini[0, 1] = 1;
a[n_, k_] := Binomial[n, k] Fubini[k, 1];
Table[a[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Mar 30 2016 *)

Formula

As infinite lower triangular matrices, A074909 * A000670 as the main diagonal and the rest zeros.

E.g.f. (exp(x) - 1)/(2 - exp(x*t)) = x + (1 + 2*t)*x^2/2! + (1 + 3*t + 9*t^2)*x^3/3! + .... Cf. A154921. - Peter Bala, Aug 31 2016

Extensions

a(36) corrected by Jean-François Alcover, Mar 30 2016