A388788 - OEIS

A388788

Array read by downward antidiagonals: T(n,k) is the number of partitions of [n], n >= 1, k >= 1, into cycles labeled with positive integers, such that the labels sum to k.

1, 1, 1, 1, 2, 2, 1, 3, 5, 6, 1, 4, 9, 17, 24, 1, 5, 14, 34, 74, 120, 1, 6, 20, 58, 159, 394, 720, 1, 7, 27, 90, 289, 893, 2484, 5040, 1, 8, 35, 131, 475, 1702, 5872, 18108, 40320, 1, 9, 44, 182, 729, 2921, 11619, 44308, 149904, 362880

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OFFSET

1,5

LINKS

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

FORMULA

Let (x)_n be the Pochhammer symbol (x+n-1)!/(x-1)!.

T(n,k) = Sum_{j=0..k} C(k-1,j-1) * |Stirling1(n,j)|.

D-finite with recurrence T(n+1,k+1) = n*T(n,k+1) + T(n+1,k) - (n-1)*T(n,k).

E.g.f.: 1/(1-x)^(y/(1-y)).

Column e.g.f.: log(1/(1-x))*hypergeom([1-k],[2],log(1-x)).

O.g.f.: hypergeom([1,y/(1-y)],[],x) = Gamma(1-y/(1-y),-1/x) / ((-x)^(y/(1-y))*exp(1/x)).

Row o.g.f.: (y/(1-y))_n.

Almost-g.f.: T(n,k) = [x^k] (1+x)^(k-1)*(x)_n.

Sequence T(n,k)*y^k (for fixed n) has centre of mass ( Sum_{k=0..n-1} 1/(y + (1-y)*k) ) / (1-y).

EXAMPLE

n\k| 1 2 3 4 5 6 7 8

---+--------------------------------------------------

1 | 1 1 1 1 1 1 1 1

2 | 1 2 3 4 5 6 7 8

3 | 2 5 9 14 20 27 35 44 (A000096)

4 | 6 17 34 58 90 131 182 244 (A023545)

5 | 24 74 159 289 475 729 1064 1494

6 | 120 394 893 1702 2921 4666 7070 10284

7 | 720 2484 5872 11619 20635 34026 53116 79470

8 |5040 18108 44308 90409 165140 279512 447168 684762

MATHEMATICA

MatrixForm[Table[Table[Sum[Binomial[k-1, j-1]*Abs[StirlingS1[n, j]], {j, 0, k}], {k, 1, 8}], {n, 1, 8}]]

PROG

(PARI) T(n, k) = sum(j=0, k, binomial(k-1, j-1)*abs(stirling(n, j, 1))); \\ Michel Marcus, Sep 22 2025

CROSSREFS

Replacing cycles with subsets yields A387299.

Triangle A271699 is contained in the transpose.

Cf. A000774 (k=2), A000096 (n=2), A023545 (n=3), A247329 (diagonal).

Sequence in context: A095788 A071944 A309495 * A080955 A340108 A340107

Adjacent sequences: A388785 A388786 A388787 * A388789 A388790 A388791

KEYWORD

nonn,tabl

AUTHOR

Natalia L. Skirrow, Sep 20 2025

STATUS

approved