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A073146 Triangle of numbers {a(n,k), n >= 0, 0<=k<=n} defined by a(0,0)=1, a(n,0)=A000670(n), a(n,n)=A000629(n), a(n,k)=a(n,k-1)+a(n-1,k-1); a(n+1,0)=Sum a(n,k), k=0..n. 4
1, 1, 2, 3, 4, 6, 13, 16, 20, 26, 75, 88, 104, 124, 150, 541, 616, 704, 808, 932, 1082, 4683, 5224, 5840, 6544, 7352, 8284, 9366, 47293, 51976, 57200, 63040, 69584, 76936, 85220, 94586, 545835, 593128, 645104, 702304, 765344, 834928, 911864 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Related to preferential arrangements of n elements (A000670) and necklaces of sets of labeled beads (A000629).

Row sums are 1, 3, 13, 75, 541... (A000670 starting from A000670(1), the second "1"). - Gary W. Adamson, May 31 2005

LINKS

Table of n, a(n) for n=0..42.

D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

FORMULA

Double-exponential generating function: sum_{n, k} a(n-k, k) x^n/n! y^k/k! = exp(y)/(2-exp(x+y)). a(n,k) = Sum_{i=n-k..n} binomial(k,i-n+k) *A000670(i). - Vladeta Jovovic, Oct 15 2006

EXAMPLE

1,

1,   2,

3,   4,   6,

13,  16,  20,  26,

75,  88,  104, 124, 150,

541, 616, 704, 808, 932, 1082, ...

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_] := Sum[Binomial[k, i-n+k] Fubini[i, 1], {i, n-k, n}];

Table[a[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Mar 30 2016, after Vladeta Jovovic *)

CROSSREFS

Cf. A000670, A000629, A011971.

Main diagonal is in A098696.

Sequence in context: A274694 A096988 A066463 * A038767 A188715 A174046

Adjacent sequences:  A073143 A073144 A073145 * A073147 A073148 A073149

KEYWORD

easy,nonn,tabl

AUTHOR

Paul D. Hanna, Jul 18 2002

STATUS

approved

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Last modified February 23 15:00 EST 2020. Contains 332166 sequences. (Running on oeis4.)