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

1,3

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;...).

Links

Table of n, a(n) for n=1..55.

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

Example

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;
...

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 *)

Crossrefs

Cf. A000670, A052882, A154921.

Sequence in context: A101486 A086606 A076112 * A122454 A189650 A083782

Adjacent sequences: A208741 A208742 A208743 * A208745 A208746 A208747

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 05 2012

Extensions

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

Status

approved