A081046 -id:A081046 - cp's OEIS frontend

A000774 a(n) = n!*(1 + Sum_{i=1..n} 1/i).

1, 2, 5, 17, 74, 394, 2484, 18108, 149904, 1389456, 14257440, 160460640, 1965444480, 26029779840, 370643938560, 5646837369600, 91657072281600, 1579093018675200, 28779361764249600, 553210247226470400, 11185850044938240000, 237335752951879680000

Offset: 0

N. J. A. Sloane

Number of {12,12*,21}-avoiding signed permutations in the hyperoctahedral group.
Let M be the n X n matrix with M( i, i ) = i+1, other entries = 1. Then a(n) = det(M); example: a(3) = 17 = det([2, 1, 1; 1, 3, 1; 1, 1, 4]). - Philippe Deléham, Jun 13 2005.
With offset 1: number of permutations of the n-set into at most two cycles. - Joerg Arndt, Jun 22 2009
A ball goes with probability 1/(k+1) from place k to a place j with j=0..k; a(n)/n! is the average number of steps from place n to place 0. - Paul Weisenhorn, Jun 03 2010
a(n) is a multiple of A025527(n). - Charles R Greathouse IV, Oct 16 2012

			(1-x)^-1 * (1 - log(1-x)) = 1 + 2*x + 5/2*x^2 + 17/6*x^3 + ...
G.f.: 1+x = 1/(1+x) + 2*x/((1+x)*(1+2*x)) + 5*x^2/((1+x)*(1+2*x)*(1+3*x)) + 17*x^3/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + 74*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)) +...

Nathaniel Johnston, Table of n, a(n) for n = 0..250
Jean-Christophe Aval, Samuele Giraudo, Théo Karaboghossian, and Adrian Tanasa, Graph operads: general construction and natural extensions of canonical operads, arXiv:1912.06563 [math.CO], 2019.
Jean-Christophe Aval, Samuele Giraudo, Théo Karaboghossian, and Adrian Tanasa, Graph insertion operads, arXiv:2002.10926 [math.CO], 2020.
Brant Jones, Katelynn D. Kochalski, Sarah Loeb, and Julia C. Walk, Strategy-indifferent games of best choice, arXiv:2107.04872 [math.CO], 2021.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv preprint arXiv:1201.1323 [math.CO], 2012.
Sergey Kitaev and Jeffrey Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012) # 12.4.7.
C. Lenormand, Arbres et permutations II, see p. 9.
Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings, arXiv:2501.00357 [math.CO], 2024. See p. 13.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1285-1332.

Cf. A000254, A000776. Same as A081046 apart from signs.

A000774 := proc(n) local i,j; j := 0; for i to n do j := j+1/i od; (j+1)*n! end;
ZL :=[S, {S = Set(Cycle(Z),3 > card)}, labelled]: seq(combstruct[count](ZL, size=n), n=1..20); # Zerinvary Lajos, Mar 25 2008
a[0]:=1: p:=1: for n from 1 to 20 do
a[n]:=n*a[n-1]+p: p:=p*n: end do: # Paul Weisenhorn, Jun 03 2010

Table[n!(1+Sum[1/i,{i,n}]),{n,0,30}] (* Harvey P. Dale, Oct 03 2011 *)

PARI
```
a(n)=n!*(1+sum(j=1,n, 1/j ));
```

{a(n)=if(n==0, 1, polcoeff(1+x-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j*x+x*O(x^n)) )), n))} /* Paul D. Hanna, Mar 01 2012 */

E.g.f.: A(x) = (1-x)^-1 * (1 - log(1-x)).
a(n+1) = (n+1)*a(n) + n!. - Jon Perry, Sep 26 2004
a(n) = A000254(n) + n!. - Mark van Hoeij, Jul 06 2010
G.f.: 1+x = Sum_{n>=0} a(n) * x^n / Product_{k=1..n+1} (1 + k*x). - Paul D. Hanna, Mar 01 2012
a(n) = Sum_{k=0..n} (k+1)*|s(n,k)|, where s(n,k) are Stirling numbers of the first kind (A008275). - Peter Luschny, Oct 16 2012
Conjecture: a(n) +(-2*n+1)*a(n-1) +(n-1)^2*a(n-2)=0. - R. J. Mathar, Nov 26 2012

A223901 Poly-Cauchy numbers of the second kind hat c_n^(-3).

Original entry on oeis.org

1, -8, 35, -161, 854, -5248, 36966, -294714, 2628600, -25963392, 281529192, -3326287848, 42546905712, -585889457328, 8643254959008, -136013600978784, 2274436197944064, -40278639752011008, 753115809287568384, -14826614346669090816, 306574242780102220800

Offset: 0

Takao Komatsu, Mar 29 2013

sign

The poly-Cauchy numbers of the second kind hat c_n^k can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))/(m+1)^k, m=0..n).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012).
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

Cf. A081046, A223899, A223902, A223904.

Cf. A222636.

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^3: k in [0..n]]: n in [0..25]];

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^3, {k, 0, n}], {n, 0, 25}]

a(n) = sum(k=0, n, stirling(n, k, 1)*(-1)^k*(k+1)^3); \\ Michel Marcus, Nov 14 2015

a(n) = Sum_{k=0..n} Stirling1(n,k) * (-1)^k * (k+1)^3.
E.g.f.: (1 - 7 * log(1 + x) + 6 * log(1 + x)^2 - log(1 + x)^3) / (1 + x). - Ilya Gutkovskiy, Aug 10 2021
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^3 * (-log(1+x))^k / k!.
a(n) = (-1)^n * Sum_{k=0..3} k! * Stirling2(4,k+1) * |Stirling1(n+1,k+1)|. (End)

A223899 Poly-Cauchy numbers of the second kind hat c_n^(-2).

Original entry on oeis.org

1, -4, 13, -51, 244, -1392, 9260, -70508, 605320, -5788008, 61021872, -703384272, 8801449344, -118828732032, 1721888828928, -26656798602240, 439110126743040, -7669109089082880, 141557837068938240, -2753560001544053760, 56299265625742848000

Offset: 0

Takao Komatsu, Mar 29 2013

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012), p. 42-53.
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

Cf. A081046, A223901, A223902, A223904.

Cf. A222627.

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^2: k in [0..n]]: n in [0..23]]; // Vincenzo Librandi, Aug 03 2013

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^2, {k, 0, n}], {n, 0, 30}]

a(n) = sum(k=0, n, stirling(n, k, 1)*(-1)^k*(k+1)^2); \\ Michel Marcus, Nov 14 2015

a(n) = Sum_{k=0..n} (-1)^k * (k+1)^2 * Stirling1(n,k).
E.g.f.: (1 - log(1 + x) * (3 - log(1 + x))) / (1 + x). - Ilya Gutkovskiy, Aug 09 2021
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^2 * (-log(1+x))^k / k!.
a(n) = (-1)^n * Sum_{k=0..2} k! * Stirling2(3,k+1) * |Stirling1(n+1,k+1)|. (End)

A223902 Poly-Cauchy numbers of the second kind hat c_n^(-4).

Original entry on oeis.org

1, -16, 97, -531, 3148, -20940, 156680, -1310840, 12166096, -124281120, 1387313520, -16813355280, 219967479744, -3090914335104, 46439677053120, -743069262651840, 12616998421804416, -226608929801923968, 4292762009479969536, -85545808260446050560, 1789078468694176410624

Offset: 0

Takao Komatsu, Mar 29 2013

sign

The poly-Cauchy numbers of the second kind hat c_n^k can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))/(m+1)^k, m=0..n).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012), p. 42-53.
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

Cf. A081046, A223899, A223901, A223904.

Cf. A222748.

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^4, {k, 0, n}], {n, 0, 30}]

a(n) = sum(k=0, n, stirling(n, k, 1)*(-1)^k*(k+1)^4); \\ Michel Marcus, Nov 14 2015

a(n) = Sum_{k=0..n} (-1)^k * (k+1)^4 * Stirling1(n,k).
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^4 * (-log(1+x))^k / k!.
E.g.f.: (1/(1+x)) * Sum_{k=0..4} Stirling2(5,k+1) * (-log(1+x))^k.
a(n) = (-1)^n * Sum_{k=0..4} k! * Stirling2(5,k+1) * |Stirling1(n+1,k+1)|. (End)

A223904 Poly-Cauchy numbers of the second kind hat c_n^(-5).

Original entry on oeis.org

1, -32, 275, -1817, 12134, -87784, 699894, -6158058, 59566464, -630057696, 7246806720, -90151868160, 1207028135520, -17314992935040, 265048030579680, -4313510679824160, 74387763047472000, -1355291635314213120, 26016022725597866880, -524865277479851360640, 11103724030717930095360

Offset: 0

Takao Komatsu, Mar 29 2013

sign

The poly-Cauchy numbers of the second kind hat c_n^k can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))/(m+1)^k, m=0..n).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012), p. 42-53.
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

Cf. A081046, A223899, A223901, A223902.

Cf. A223023.

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^5: k in [0..n]]: n in [0..23]]; // Vincenzo Librandi, Aug 03 2013

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^5, {k, 0, n}], {n, 0, 30}]

a(n) = sum(k=0, n, (-1)^k*stirling(n, k, 1)*(k+1)^5); \\ Michel Marcus, Nov 14 2015

a(n) = Sum_{k=0..n} (-1)^k * (k+1)^5 * Stirling1(n,k).
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^5 * (-log(1+x))^k / k!.
E.g.f.: (1/(1+x)) * Sum_{k=0..5} Stirling2(6,k+1) * (-log(1+x))^k.
a(n) = (-1)^n * Sum_{k=0..5} k! * Stirling2(6,k+1) * |Stirling1(n+1,k+1)|. (End)

A081047 Difference of Stirling numbers of the first kind.

Original entry on oeis.org

1, 0, -1, -5, -26, -154, -1044, -8028, -69264, -663696, -6999840, -80627040, -1007441280, -13575738240, -196287356160, -3031488633600, -49811492505600, -867718162483200, -15974614352793600, -309920046408806400, -6320046028584960000

Offset: 0

Paul Barry, Mar 05 2003

G. C. Greubel, Table of n, a(n) for n = 0..400
Thierry Dana-Picard and David G. Zeitoun, Sequences of definite integrals, infinite series and Stirling numbers, International Journal of Mathematical Education in Science and Technology, Volume 43, 2012 - Issue 2.
Motohico Mulase, In Search of a Hidden Curve, arXiv:2501.00716 [math.QA], 2025. See p. 27.

Cf. A001705, A008275, A081046.

With[{nn = 100}, CoefficientList[Series[(1 + Log[1 - x])/(1 - x), {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, Jan 21 2017 *)

E.g.f.: (1+log(1-x))/(1-x). - Paul Barry, Nov 26 2008
a(n) = abs(s(n+1, 1))-abs(s(n+1, 2)), where s(n, m) is a (signed) Stirling number of the first kind (A008275). (corrected by Wolfdieter Lang, Jun 20 2011)
a(n) = A094645(n+2,2), n>=0. - _Wolfdieter Lang, Jun 20 2011

A096747 Triangle read by rows: T(n,1) = 1, T(n,k) = T(n-1,k)+(n-1)*T(n-1,k-1) for 1<=k<=n+1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 7, 18, 24, 24, 1, 11, 46, 96, 120, 120, 1, 16, 101, 326, 600, 720, 720, 1, 22, 197, 932, 2556, 4320, 5040, 5040, 1, 29, 351, 2311, 9080, 22212, 35280, 40320, 40320, 1, 37, 583, 5119, 27568, 94852, 212976, 322560, 362880

Offset: 0

Thomas J Engelsma (tom(AT)opertech.com), Dec 05 2004

Note: rows continue as factorials - stopped at second factorial for clarity.
T(n,n) = T(n,n+1) = n!. Sum of row n = n! + s(n,2), where s(n,2) are signless Stirling numbers of the first kind (A081046). T(n,k) = A109822(n,k) for 1<=k<=n (i.e. triangle without the last column is A109822). - Emeric Deutsch, Jul 03 2005
Sum(k=0..n-1, T(n,k))/T(n,n-1) are for n>=1 the harmonic numbers A001008(n)/A002805(n). - Peter Luschny, Sep 15 2014

			Triangle begins:
*0.........................1
*1......................1.....1
*2...................1.....2.....2
*3................1.....4.....6.....6
*4.............1.....7....18....24....24
*5..........1....11....46....96...120...120
*6.......1....16...101...326...600...720...720
*7....1....22...197...932..2556..4320..5040..5040
T(5,3)=46 because 4*7+18=46

Robert Israel, Table of n, a(n) for n = 0..10152 (rows 0..141, flattened)
R. P. Stanley, Ordering events in Minkowski space, arXiv:math/0501256 [math.CO], 2005.

Cf. A081046, A109822.

T:=proc(n,k) if k=1 then 1 elif k=n+1 then n! else T(n-1,k)+(n-1)*T(n-1,k-1) fi end: for n from 0 to 11 do seq(T(n,k),k=1..n+1) od; # yields sequence in triangular form
with(combinat): T:=(n,k)->sum(abs(stirling1(n,n-i)),i=0..k-1): for n from 0 to 11 do seq(T(n,k),k=1..n+1) od; # yields sequence in triangular form; Emeric Deutsch, Jul 03 2005

T[n_, k_] := Sum[Abs[StirlingS1[n, n - i]], {i, 0, k}]; T[0, 0] := 1;
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 08 2016 *)

@CachedFunction
def T(n,k):
    if n == 0: return 1
    if k < 0: return 0
    return T(n-1,k)+(n-1)*T(n-1,k-1)
for n in range(9): print([T(n,k) for k in (0..n)]) # Peter Luschny, Sep 15 2014

T(n+1, i) = n*T(n, i-1)+T(n, i)
T(n, k) = sum(|stirling1(n, n-i)|, i=0..k-1) for 1<=k<=n. - Emeric Deutsch, Jul 03 2005
E.g.f. as triangle: g(x,y) = Sum_{n>=0} Sum_{1<=k<=n+1} T(n,k) x^n y^k/n! where
g(x,y) = -y^2/((y-1)*(x*y-1)) - (1-x*y)^(-1/y)*(-y+y^2/(y-1)). - Robert Israel, Nov 28 2016

More terms from Emeric Deutsch, Jul 03 2005

A349782 Triangle read by rows, T(n, k) = Sum_{j=0..k} |Stirling1(n, j)|.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 5, 6, 0, 6, 17, 23, 24, 0, 24, 74, 109, 119, 120, 0, 120, 394, 619, 704, 719, 720, 0, 720, 2484, 4108, 4843, 5018, 5039, 5040, 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320, 0, 40320, 149904, 268028, 335312, 357761, 362297, 362843, 362879, 362880

Offset: 0

Peter Luschny, Dec 02 2021

T(n, k) is the number of permutations of n objects that contain at most k cycles.

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,    2;
[3] 0, 2,    5,     6;
[4] 0, 6,    17,    23,    24;
[5] 0, 24,   74,    109,   119,   120;
[6] 0, 120,  394,   619,   704,   719,   720;
[7] 0, 720,  2484,  4108,  4843,  5018,  5039,  5040;
[8] 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320;

Row sums: A121586, central terms: A349783.

Cf. A132393, A048994, A008275, A000142, A033312, A000774, A081046, A130534.

T := (n, k) -> add(abs(Stirling1(n,j)), j = 0..k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

T[n_, k_] := Sum[Abs[StirlingS1[n, j]], {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 09 2021 *)

T(n, k) = sum(j=0, k, abs(stirling(n, j, 1))); \\ Michel Marcus, Dec 09 2021

T(n,k) = Sum_{j=0..k} A132393(n,j). - Alois P. Heinz, Dec 10 2021

A081103 Alternating sum of first three Stirling numbers of the first kind.

Original entry on oeis.org

0, 1, -4, 18, -95, 584, -4123, 32969, -294992, 2922956, -31791716, 376719892, -4832017320, 66713229192, -986611705584, 15561976320144, -260804276106624, 4628322010931328, -86710491660063744, 1710290952899283456, -35427639035553292800, 768970029545198092800

Offset: 0

Paul Barry, Mar 05 2003

Cf. A000399, A081052, A081046.

a(n) = s(n, 1)-s(n, 2)+s(n, 3), s(n, m)= signed Stirling numbers of the first kind.

E.g.f.: (1+x)^(-1)*(log(1+x)-log(1+x)^2/2+log(1+x)^3/6).

cp's OEIS Frontend

A000774 a(n) = n!*(1 + Sum_{i=1..n} 1/i).

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A223901 Poly-Cauchy numbers of the second kind hat c_n^(-3).

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A223899 Poly-Cauchy numbers of the second kind hat c_n^(-2).

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A223902 Poly-Cauchy numbers of the second kind hat c_n^(-4).

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A223904 Poly-Cauchy numbers of the second kind hat c_n^(-5).

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A081047 Difference of Stirling numbers of the first kind.

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A096747 Triangle read by rows: T(n,1) = 1, T(n,k) = T(n-1,k)+(n-1)*T(n-1,k-1) for 1<=k<=n+1.

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