A208744 - OEIS

%I #20 Aug 31 2016 05:46:11

%S 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,

%T 2625,11361,32781,1,8,84,728,5250,30296,131124,378344,1,9,108,1092,

%U 9450,68166,393372,1702548,4912515,1,10,135,1560,15750,136332,983430,5675160,24562575,70872610

%N Triangle relating to ordered Bell numbers, A000670.

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

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

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

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

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

%F 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

%e Row 4 (nonzero terms) = (1, 4, 18, 52) = termwise product of (1, 4, 6, 4) and (1, 1, 3, 13).

%e First few rows of the triangle:

%e 1;

%e 1, 2;

%e 1, 3, 9;

%e 1, 4, 18, 52;

%e 1, 5, 30, 130, 375;

%e 1, 6, 45, 260, 1125, 3246;

%e 1, 7, 63, 455, 2625, 11361, 32781;

%e 1, 8, 84, 728, 5250, 30296, 131124, 378344;

%e ...

%t 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;

%t a[n_, k_] := Binomial[n, k] Fubini[k, 1];

%t Table[a[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, Mar 30 2016 *)

%Y Cf. A000670, A052882, A154921.

%K nonn,tabl

%O 1,3

%A _Gary W. Adamson_, Mar 05 2012

%E a(36) corrected by _Jean-François Alcover_, Mar 30 2016