0,7

The k-th column sequence, k >= 0, without leading zeros, enumerates the ways to distribute n beads, n >= 1, labeled differently from 1 to n, over a set of (unordered) necklaces, excluding necklaces with exactly one bead, and k+1 indistinguishable, ordered, fixed cords, each allowed to have any number of beads. Beadless necklaces as well as beadless cords each contribute a factor 1, hence for n=0 one has 1. See A000255 for the description of a fixed cord with beads. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 02 2010

Indranil Ghosh, Rows 0..50, flattened

W. Y. C. Chen et al., Higher-order log-concavity in Euler's difference table, Discrete Math., 311 (2011), 2128-2134. (These are the numbers d^k_n.)

Fanja Rakotondrajao, k-Fixed-Points-Permutations, Integers: Electronic journal of combinatorial number theory 7 (2007) A36.

T(n, n) = 1; T(n+1, n) = n.

T(n+2, n) = A002061(n+1) = n^2 + n + 1; T(n+3, n) = n^3 + 3*n^2 + 5*n + 2.

T(n, k) = (k + 1)*T(n, k + 1) - T(n-1, k); T(n, n) = 1; T(n, k) = 0, if k > n.

T(n, k) = (n-1)*T(n-1, k) + (n-k-1)*T(n-2, k).

k!*T(n, k) = A068106(n, k). [corrected by Georg Fischer, Aug 13 2022]

Sum_{k>=0} T(n, k) = A003470(n+1).

T(n, k) = (1/k!) * Sum_{j>=0} (-1)^j*binomial(n-k, j)*(n-j)!. - Philippe Deléham, Jun 13 2005

From Peter Bala, Aug 14 2008: (Start)

The following remarks all relate to the array read as a square array: e.g.f for column k: exp(-y)/(1-y)^(k+1); e.g.f. for array: exp(-y)/(1-x-y) = (1 + x + x^2 + x^3 + ...) + (x + 2*x^2 + 3*x^3 + 4*x^4 + ...)*y + (1 + 3*x + 7*x^2 + 13*x^3 + ...)*y^2/2! + ... .

This table is closely connected to the constant e. The row, column and diagonal entries of this table occur in series formulas for e.

Row n for n >= 2: e = n!*(1/T(n,0) + (-1)^n*[1/(1!*T(n,0)*T(n,1)) + 1/(2!*T(n,1)*T(n,2)) + 1/(3!*T(n,2)*T(n,3)) + ...]). For example, row 3 gives e = 6*(1/2 - 1/(1!*2*11) - 1/(2!*11*32) - 1/(3!*32*71) - ...). See A095000.

Column 0: e = 2 + Sum_{n>=2} (-1)^n*n!/(T(n,0)*T(n+1,0)) = 2 + 2!/(1*2) - 3 !/(2*9) + 4!/(9*44) - ... .

Column k, k >= 1: e = (1 + 1/1! + 1/2! + ... + 1/k!) + 1/k!*Sum_{n >= 0} (-1)^n*n!/(T(n,k)*T(n+1,k)). For example, column 3 gives e = 8/3 + 1/6*(1/(1*3) - 1/(3*13) + 2/(13*71) - 6/(71*465) + ...).

Main diagonal: e = 1 + 2*(1/(1*1) - 1/(1*7) + 1/(7*71) - 1/(71*1001) + ...).

First subdiagonal: e = 8/3 + 5/(3*32) - 7/(32*465) + 9/(465*8544) - ... .

Second subdiagonal: e = 2*(1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...). See A143413.

Third subdiagonal: e = 3 - (2*3*5)/(2*53) + (3*4*7)/(53*1214) - (4*5*9)/(1214*30637) + ... .

For the corresponding results for the constants 1/e, sqrt(e) and 1/sqrt(e) see A143409, A143410 and A143411 respectively. For other arrays similarly related to constants see A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)). (End)

G.f. for column k is hypergeom([1,k+1],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011

T(n, k) = (n!/k!)*hypergeom([k-n], [-n], -1). - Peter Luschny, Oct 05 2017

Formatted as a square array:

1 3 7 13 21 31 43 57 ... A002061;

2 11 32 71 134 227 356 ... A094792;

9 53 181 465 1001 1909 ... A094793;

44 309 1214 3539 8544 ... A094794;

265 2119 9403 30637 ... A023043;

1854 16687 82508 ... A023044;

14833 148329 ... A023045;

Formatted as a triangular array (mirror of A076731):

1;

0 1;

1 1 1;

2 3 2 1;

9 11 7 3 1;

44 53 32 13 4 1;

265 309 181 71 21 5 1;

1854 2119 1214 465 134 31 6 1;

14833 16687 9403 3539 1001 227 43 7 1;

133496 148329 82508 30637 8544 1909 356 57 8 1;

T[n_, k_]:=(1/k!)*Sum[(-1)^j*Binomial[n-k, j]*(n-j)!, {j, 0, n}]; Flatten[Table[T[n, k], {n, 0, 11}, {k, 0, n}]] (* Indranil Ghosh, Feb 20 2017 *)

T[n_, k_] := (n!/k!) HypergeometricPFQ[{k-n}, {-n}, -1];

Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Peter Luschny, Oct 05 2017 *)

(Magma)

A086764:= func< n, k | (&+[(-1)^j*Binomial(n-k, j)*Factorial(n-j): j in [0..n]])/Factorial(k) >;

[A086764(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

(SageMath)

def A086764(n, k): return sum((-1)^j*binomial(n-k, j)*factorial(n-j) for j in range(n+1))//factorial(k)

flatten([[A086764(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

Columns: A000166, A000155, A000153, A000261, A001909, A001910, A176732, A176733, A176734, A176735, A176736.

Mirror image of A076731.

Cf. A002061, A003470, A008288, A023043, A023044, A023045, A068106.

Cf. A076731, A094792, A094793, A094794, A095000, A108625, A143007.

Cf. A143409, A143410, A143411, A143413.

Sequence in context: A291087 A020858 A090664 * A375089 A255010 A292371

Adjacent sequences: A086761 A086762 A086763 * A086765 A086766 A086767

easy,nonn,tabl

Philippe Deléham, Aug 02 2003

More terms from David Wasserman, Mar 28 2005

Additional comments from Zerinvary Lajos, Mar 30 2006

Edited by N. J. A. Sloane, Sep 24 2011

approved