A259456 -id:A259456 - cp's OEIS frontend

A000276 Associated Stirling numbers.

3, 20, 130, 924, 7308, 64224, 623376, 6636960, 76998240, 967524480, 13096736640, 190060335360, 2944310342400, 48503818137600, 846795372595200, 15618926924697600, 303517672703078400, 6198400928176128000, 132720966600284160000, 2973385109386137600000

Offset: 4

N. J. A. Sloane

nonn

a(n) is also the number of permutations of n elements, without any fixed point, with exactly two cycles. - Shanzhen Gao, Sep 15 2010

			a(4) = 3 because we have: (12)(34),(13)(24),(14)(23). - _Geoffrey Critzer_, Nov 03 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Shanzhen Gao, Permutations with Restricted Structure (in preparation).

Alois P. Heinz, Table of n, a(n) for n = 4..150

A diagonal of triangle in A008306.

Cf. A052518, A052881, A259456.

nn=25;a=Log[1/(1-x)]-x;Drop[Range[0,nn]!CoefficientList[Series[a^2/2,{x,0,nn}],x],4]  (* Geoffrey Critzer, Nov 03 2012 *)
a[n_] := (n-1)!*(HarmonicNumber[n-2]-1); Table[a[n], {n, 4, 23}] (* Jean-François Alcover, Feb 06 2016, after Gary Detlefs *)

a(n) = (n-1)!*sum(i=2, n-2, 1/i); \\ Michel Marcus, Feb 06 2016

a(n) = (n-1)!*Sum_{i=2..n-2} 1/i = (n-1)!*(Psi(n-1)+gamma-1). - Vladeta Jovovic, Aug 19 2003
With alternating signs: Ramanujan polynomials psi_3(n-2, x) evaluated at 1. - Ralf Stephan, Apr 16 2004
E.g.f.: ((x+log(1-x))^2)/2. [Corrected by Vladeta Jovovic, May 03 2008]
a(n) = Sum_{i=2..floor((n-1)/2)} n!/((n-i)*i) + Sum_{i=ceiling(n/2)..floor(n/2)} n!/(2*(n-i)*i). - Shanzhen Gao, Sep 15 2010
a(n) = (n+3)!*(h(n+2)-1), with offset 0, where h(n)=sum(1/k,k=1..n). - Gary Detlefs, Sep 11 2010
Conjecture: (-n+2)*a(n) +(n-1)*(2*n-5)*a(n-1) -(n-1)*(n-2)*(n-3)*a(n-2)=0. - R. J. Mathar, Jul 18 2015
Conjecture: a(n) +2*(-n+2)*a(n-1) +(n^2-6*n+10)*a(n-2) +(n-3)*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 18 2015
a(n) = A000254(n-1) - (n-1)! - (n-2)!. - Anton Zakharov, Sep 24 2016

More terms from Christian G. Bower

A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)binomial(n + k, n + m) |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 6, 20, 15, 0, 24, 130, 210, 105, 0, 120, 924, 2380, 2520, 945, 0, 720, 7308, 26432, 44100, 34650, 10395, 0, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 0, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025

Offset: 0

Peter Luschny, Mar 27 2016

We propose to call this sequence the 'Ward cycle numbers' and sequence A269939 the 'Ward set numbers'. - Peter Luschny, Nov 25 2022

			Triangle T(n,k) starts:
  [1]
  [0,   1]
  [0,   2,      3]
  [0,   6,     20,     15]
  [0,  24,    130,    210,    105]
  [0,  120,   924,   2380,   2520,    945]
  [0,  720,  7308,  26432,  44100,  34650,  10395]
  [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135]

Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
Peter Luschny, The P-transform.
Anthony Mansuy, Preordered forests, packed words and contraction algebras, J. Algebra 411 (2014) 259-311, section 4.4.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
Nico M. Temme, Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters, Integral Transforms and Special Functions, 2021.
Nico M. Temme and Ed J. M. Veling, Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z, arXiv:2202.12857 [math.CA], 2022.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Variants: A111999, A259456.

Cf. A269939 (Stirling2 counterpart), A268438, A032188 (row sums).

T := (n, k) -> add((-1)^(m+k)*binomial(n+k,n+m)*abs(Stirling1(n+m, m)), m=0..k):
seq(print(seq(T(n, k), k=0..n)), n=0..6);
# Alternatively:
T := proc(n, k) option remember;
    `if`(k=0, k^n,
    `if`(k<=0 or k>n, 0,
    (n+k-1)*(T(n-1, k)+T(n-1, k-1)))) end:
for n from 0 to 6 do seq(T(n, k), k=0..n) od;

T[n_, k_] := Sum[(-1)^(m+k)*Binomial[n+k, n+m]*Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 12 2022 *)

T = lambda n, k: sum((-1)^(m+k)*binomial(n+k, n+m)*stirling_number1(n+m, m) for m in (0..k))
for n in (0..7): print([T(n, k) for k in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: n/(n+1), lambda n, k: (-1)^k*falling_factorial(n+k,n))

T(n,k) = (-1)^k*FF(n+k,n)*P[n,k](n/(n+1)) where P is the P-transform and FF the falling factorial function. For the definition of the P-transform see the link.

T(n,k) = A268438(n,k)*FF(n+k,n)/(2*n)!.

Name corrected after notice from Ed Veling by Peter Luschny, Jun 14 2022

A213953 Triangle by rows, inverse of A208891.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, -2, 5, 0, -3, -1, 1, -9, 5, 10, -2, -4, -1, 1, -9, -21, 25, 15, -5, -5, -1, 1, 50, -105, -11, 62, 19, -9, -6, -1, 1, 267, -141, -301, 56, 119, 21, -14, -7, -1, 1, 413, 777

Offset: 0

Gary W. Adamson, Jun 26 2012

			Triangle starts:
1;
-1, 1
0, -1, 1
1, -1, -1, 1;
1, 1, -2, -1, 1;
-2, 5, 0, -3, -1, 1;
-9, 5, 10, -2, -4, -1, 1;
-9, -21, 25, 15, -5, -5, -1, 1;
50, -105, -11, 62, 19, -9, -6, -1, 1;
267, -141, -301, 56, 119, 21, -14, -7, -1, 1;
413, 777, -1040, -566, 226, 198, 20, -20, -8, -1, 1;
...

Cf. A208891, A000587 (first column), A014619 (2nd column), A080956 (4th subdiagonal).

A208891 := proc(n,k)
    if n <0 or k<0 or k>n then
            0;
    elif n = k then
            1 ;
    else
            binomial(n-1,k) ;
    end if;
end proc:
A259456 := proc(n)
    local A, row, col ;
    A := Matrix(n, n) ;
    for row from 1 to n do
        for col from 1 to n do
            A[row, col] := A208891(row-1,col-1) ;
        end do:
    end do:
    LinearAlgebra[MatrixInverse](A) ;
end proc:
A259456(20) ; # R. J. Mathar, Jul 21 2015

Inverse of triangle A208891, Pascal's triangle matrix with an appended right border of 1's.

cp's OEIS Frontend

A000276 Associated Stirling numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)binomial(n + k, n + m) |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Sage

Formula

Extensions

A213953 Triangle by rows, inverse of A208891.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

cp's OEIS Frontend

A000276 Associated Stirling numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)*binomial(n + k, n + m) * |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Sage

Formula

Extensions

A213953 Triangle by rows, inverse of A208891.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)binomial(n + k, n + m) |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.