A008306 -id:A008306 - cp's OEIS frontend

A345697 Expansion of the e.g.f. sqrt(1 / (2exp(x) - 2x*exp(x) - 1)).

1, 0, 1, 2, 12, 64, 485, 4038, 39991, 441992, 5492322, 75171700, 1127989577, 18381446004, 323527186957, 6114296752718, 123513004310640, 2655648779976640, 60554669008300565, 1459559515622280282, 37079264125376670955, 990226180225789628660, 27733277682719819190246, 812818183963966524137332, 24880254143735238825011057

Offset: 0

Mélika Tebni, Jun 24 2021

nonn

			sqrt(1/(2*exp(x)-2*x*exp(x)-1)) = 1 + x^2/2! + 2*x^3/3! + 12*x^4/4! + 64*x^5/5! + 485*x^6/6! + 4038*x^7/7! + 39991*x^8/8! + 441992*x^9/9! + ...
a(13) = Sum_{k=1..6} A014307(k)*A008306(13,k) = 18381446004.
A014307(1)*A008306(13,1) == -1 (mod 13), because A014307(1) = 1 and A008306(13,1) = (13-1)!
For k>=2, A008306(13,k) == 0 (mod 13), result a(13) == -1 (mod 13).

Cf. A000166, A008306, A014307, A327006.

A014307 := proc(n) option remember; `if`(n=0, 1 , 1+add((-1+binomial(n, k))*A014307(k), k=1..n-1)) end:
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a := n-> add((A014307(k)*A008306(n,k)), k=1..floor(n/2)):a(0):=1 ;
seq(a(n), n=0..24);
# second program:
a := series(sqrt((1/(2*exp(x)-2*x*exp(x)-1))), x=0, 25):
seq(n!*coeff(a, x, n), n=0..24);

CoefficientList[Series[Sqrt[1/(2*E^x-2*x*E^x-1)], {x, 0, 24}], x] * Range[0, 24]!

my(x='x+O('x^25)); Vec(serlaplace(sqrt(1 / (2*exp(x) - 2*x*exp(x) -1)))) \\ Michel Marcus, Jun 24 2021

E.g.f. y(x) satisfies y' = x*exp(x)*y^3.
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} A014307(k)*A008306(n,k) for n >= 1.
For all p prime, a(p) == -1 (mod p).
a(n) ~ sqrt(2*c) * n^n / ((1-c)^(n+1) * exp(n)), where c = -LambertW(-exp(-1)/2). - Vaclav Kotesovec, Jun 25 2021

A345969 Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x))).

Original entry on oeis.org

1, 0, 1, 2, 18, 104, 1015, 9666, 116557, 1504856, 22300704, 358916480, 6373675825, 122332173300, 2540560235161, 56558354414870, 1346402030278050, 34093192112537888, 915570658175517151, 25983157665663651150, 777141557158947654637, 24430880483991543481580

Offset: 0

Mélika Tebni, Jul 01 2021

nonn

			1/sqrt(3-2/((1-x)*exp(x))) =  1 + x^2/2! + 2*x^3/3! + 18*x^4/4! + 104*x^5/5! + 1015*x^6/6! + 9666*x^7/7! + 116557*x^8/8! + 1504856*x^9/9! + ...
a(17) = Sum_{k=1..8} A305404(k)*A008306(17,k) = 34093192112537888.
For k=1, A305404(1)*A008306(17,1) == -1 (mod 17), because A305404(1) = 1 and A008306(17,1) = (17-1)!
For k>=2, A305404(k)*A008306(17,k) == 0 (mod 17), because A008306(17,k) == 0 (mod 17), result a(17) == -1 (mod 17).

Cf. A000166, A008306, A305404, A327006, A345697.

A305404:= n-> add(Stirling2(n,k)*doublefactorial(2*k-1), k=0..n):
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a := n-> add((A305404(k)*A008306(n, k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(1/sqrt(3-2/((1-x)*exp(x))), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);

CoefficientList[Series[1/Sqrt[3-2/((1-x)*E^x)], {x, 0, 24}], x] * Range[0, 24]!

my(x='x+O('x^30)); Vec(serlaplace(1/sqrt(3 - 2 / ((1 - x)*exp(x))))) \\ Michel Marcus, Jul 01 2021

E.g.f. y(x) satisfies y' = exp(-x)*y^3*x/(1-x)^2.
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} A305404(k)*A008306(n,k) for n > 0.
For all p prime, a(p) == -1 (mod p).
a(n) ~ sqrt(-2*LambertW(-2*exp(-1)/3)/3) * n^n / (exp(n) * (1 + LambertW(-2*exp(-1)/3))^(n+1)). - Vaclav Kotesovec, Jul 01 2021

A050211 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=3.

Original entry on oeis.org

2, 6, 24, 120, 40, 720, 420, 5040, 3948, 40320, 38304, 2240, 362880, 396576, 50400, 3628800, 4419360, 859320, 39916800, 53048160, 13665960, 246400, 479001600, 684478080, 216339552, 9609600, 6227020800, 9464307840, 3501834336

Offset: 3

Eric W. Weisstein

Generalizes Stirling numbers of the first kind

			Table begins:
   n\k |      u     u^2    u^3
  = = = = = = = = = = = = = = =
    3  |      2
    4  |      6
    5  |     24
    6  |    120     40
    7  |    720    420
    8  |   5040   3948
    9  |  40320  38304    2240
   ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.

Alois P. Heinz, Rows n = 3..200, flattened
S. Brassesco, M. A. Méndez, The asymptotic expansion for the factorial and Lagrange inversion formula, arXiv:1002.3894v1 [math.CA], 2010.
G. Nemes, On the Coefficients of the Asymptotic Expansion of n!, J. Int. Seq. 13 (2010), 10.6.6.
Eric Weisstein's World of Mathematics, Permutation Cycle.

Cf. A008275, A008306, A050212, A050213.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=3..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=3..15);  # Alois P. Heinz, Sep 25 2016

t[n_ /; n >= 3, k_ /; k >= 1] := t[n, k] = (n - 1)*t[n - 1, k] + (n - 2)*(n - 1)*t[n - 3, k - 1] ; t[, ] = 0; t[3, 1] = 2; Flatten[ Table[t[n, k], {n, 3, 15}, {k, 1, Floor[n/3]}]] (* Jean-François Alcover, Nov 05 2012, after Peter Bala *)

From Peter Bala, Sep 06 2011: (Start)
E.g.f.: (1-t)^(-u)*exp(-u*(t+t^2/2)) - 1 = (2*u)*t^3/3!+(6*u)*t^4/4!+(24*u)*t^5/5!+(120*u+40*u^2)*t^6/6!+(720*u+420*u^2)*t^7/7!+....
E.g.f. for column k: 1/k!*(-log(1-x)-x-x^2/2)^k.
Starting at row 3, row lengths are 1, 1, 1, 2, 2, 2, 3, 3, 3, ....
Recurrence: T(n,k) = (n-1)*T(n-1,k) + (n-1)*(n-2)*T(n-3,k-1). (End)

A050212 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=4.

Original entry on oeis.org

6, 24, 120, 720, 5040, 1260, 40320, 18144, 362880, 223776, 3628800, 2756160, 39916800, 35307360, 1247400, 479001600, 476910720, 38918880, 6227020800, 6822541440, 889945056, 87178291200, 103440879360, 18478684224

Offset: 4

Eric W. Weisstein

Generalizes Stirling numbers of the first kind.

			Triangle begins:
:        6;
:       24;
:      120;
:      720;
:     5040,     1260;
:    40320,    18144;
:   362880,   223776;
:  3628800,  2756160;
: 39916800, 35307360, 1247400;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.

Alois P. Heinz, Rows n = 4..300, flattened
Eric Weisstein's World of Mathematics, Permutation cycle.

Cf. A008275, A008306, A050211, A050213.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=4..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=4..20);  # Alois P. Heinz, Sep 25 2016

b[n_] := b[n] = Expand[If[n==0, 1, Sum[b[n-i] x Binomial[n-1, i-1] (i-1)!, {i, 4, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x]}]][b[n]];
Table[T[n], {n, 4, 20}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Offset changed from 1 to 4 by Alois P. Heinz, Sep 25 2016

A050213 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=5.

Original entry on oeis.org

24, 120, 720, 5040, 40320, 362880, 72576, 3628800, 1330560, 39916800, 20338560, 479001600, 303937920, 6227020800, 4643084160, 87178291200, 73721007360, 1743565824, 1307674368000, 1224694598400, 69742632960, 20922789888000

Offset: 5

Eric W. Weisstein

Generalizes Stirling numbers of the first kind.

			Triangle begins:
05:       24;
06:      120;
07:      720;
08:     5040;
09:    40320;
10:   362880,    72576;
11:  3628800,  1330560;
12: 39916800, 20338560;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.

Alois P. Heinz, Rows n = 5..300, flattened
Eric Weisstein's World of Mathematics, Permutation Cycle.

Cf. A008275, A008306, A050211, A050212.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=5..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=5..20);  # Alois P. Heinz, Sep 25 2016

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*x*Binomial[n - 1, i - 1]* (i - 1)!, {i, 5, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n]];
T /@ Range[5, 20] // Flatten (* Jean-François Alcover, Dec 08 2019, after Alois P. Heinz *)

Offset changed from 1 to 5 by Alois P. Heinz, Sep 25 2016

A079510 Triangle T(n,k) read by rows; related to number of preorders.

Original entry on oeis.org

1, 0, 2, 0, 3, 6, 0, 0, 20, 24, 0, 0, 15, 130, 120, 0, 0, 0, 210, 924, 720, 0, 0, 0, 105, 2380, 7308, 5040, 0, 0, 0, 0, 2520, 26432, 64224, 40320, 0, 0, 0, 0, 945, 44100, 303660, 623376, 362880, 0, 0, 0, 0, 0, 34650, 705320, 3678840, 6636960, 3628800

Offset: 1

N. J. A. Sloane, Jan 21 2003

There are only m nonzero entries in the m-th column.

			Triangle begins:
  1;
  0,   2;
  0,   3,   6;
  0,   0,  20,  24;
  0,   0,  15, 130, 120;
  ...

G. C. Greubel, Rows n=1..30 of triangle, flattened
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (See the array on page 29.)
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)

A rearrangement of the triangle in A008306. - Benoit Cloitre, Jan 27 2003

T[n_, k_]:= If[k < 1 || k > n, 0, If[n==1 && k==1, 1, n*(T[n-1, k-1] + T[n-2, k-1])]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Jan 17 2019 *)

T(n,k)=if(k<=0 || k>n, 0, if(n==1 && k==1, 1, n*(T(n-1,k-1)+T(n-2,k-1))));

Recurrence and more terms from Michael Somos, Jan 23 2003

Offset changed to 1 by G. C. Greubel, Jan 17 2019

A346119 Expansion of the e.g.f. sqrt(2xexp(x) - 2*exp(x) + 3).

Original entry on oeis.org

1, 0, 1, 2, 0, -16, -35, 342, 2779, -6424, -239382, -822460, 22393657, 278844084, -1553468891, -68399947042, -275025888900, 15302175612416, 243541868882077, -2463105309082902, -121649966081262521, -473088821582805820, 50905612811064360006, 945133249101683013812, -15321255878414345388335

Offset: 0

Mélika Tebni, Jul 05 2021

sign

			sqrt(2*x*exp(x)-2*exp(x)+3) = 1 + x^2/2! + 2*x^3/3! - 16*x^5/5! - 35*x^6/6! + 342*x^7/7! + 2779*x^8/8! - 6424*x^9/9! + ...
a(11) = Sum_{k=1..5} (-1)^(k-1)*A006677(k)*A008306(11,k) = -822460.
For k=1, (-1)^(1-1)*A006677(1)*A008306(11,1) == -1 (mod 11), because A006677(1) = 1 and A008306(11,1) = (11-1)!
For k>=2, (-1)^(k-1)*A006677(k)*A008306(11,k) == 0 (mod 11), because A008306(11,k) == 0 (mod 11), result a(11) == -1 (mod 11).
a(8) = Sum_{k=1..4} (-1)^(k-1)*A006677(k)*A008306(8,k) = 2779.
a(8) == 0 (mod (8-1)), because for k >= 1, A008306(8,k) == 0 (mod 7).

Cf. A000166, A006677, A008306, A327006, A345652, A345697, A345969.

stirtr:= proc(p) proc(n) add(p(k)*Stirling2(n, k), k=0..n) end end: f:= n-> `if`(n=0, 1, (2*n-2)!/ (n-1)!/ 2^(n-1)): A006677:= stirtr(f): # Alois P. Heinz, 2008.
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a:= n-> add(((-1)^(k-1)*A006677(k)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(sqrt(2*x*exp(x)-2*exp(x)+3), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);

CoefficientList[Series[Sqrt(2*x*E^x-2*E^x+3), {x, 0, 24}], x] * Range[0, 24]!

my(x='x+O('x^30)); Vec(serlaplace(sqrt(2*x*exp(x) - 2*exp(x) + 3))) \\ Michel Marcus, Jul 05 2021

E.g.f. y(x) satisfies y*y' = x*exp(x).
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1)*A006677(k)*A008306(n,k) for n > 0.
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
Conjecture: a(n) = 0 for only n = 1 and n = 4.

A162973 Number of cycles in all derangement permutations of {1,2,...,n}.

Original entry on oeis.org

0, 1, 2, 12, 64, 425, 3198, 27216, 258144, 2701737, 30933770, 384675148, 5163521856, 74417353985, 1146203362822, 18790377267840, 326682354342336, 6003886529652657, 116305541572943826, 2368629865508978284

Offset: 1

Emeric Deutsch, Jul 22 2009

nonn

			a(4)=12 because in the derangements of {1,2,3,4}, namely (12)(34), (13)(24), (14)(23), (1234), (1243), (1324), (1342), (1423), and (1432), we have a total of 2+2+2+1+1+1+1+1+1=12 cycles.

G. C. Greubel, Table of n, a(n) for n = 1..449
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.

Cf. A008306, A162972, A000254.

G := exp(-z)*(z+ln(1-z))/(z-1): Gser := series(G, z = 0, 25): seq(factorial(n)*coeff(Gser, z, n), n = 1 .. 22);

With[{nn=20},Rest[CoefficientList[Series[Exp[-x] (x+Log[1-x])/(x-1), {x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Jul 25 2013 *)

x='x+O('x^30); concat([0], Vec(serlaplace(exp(-x)*(x+log(1-x))/(x -1)))) \\ G. C. Greubel, Sep 01 2018

a(n) = Sum_{k>=1} k*A008306(n,k).
E.g.f.: exp(-z)*(z+log(1-z))/(z-1).
a(n) ~ n! * (log(n) + gamma - 1)/exp(1), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 25 2013
a(n) = A000254(n) - A162972(n). - Anton Zakharov, Oct 18 2016
D-finite with recurrence a(n) +2*(-n+2)*a(n-1) +(n-2)*(n-6)*a(n-2) +(3*n-8)*(n-3)*a(n-3) +3*(n-3)^2*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

A343482 Expansion of the e.g.f. sqrt(-1 + 2 / (1 - x) / exp(x)).

Original entry on oeis.org

1, 0, 1, 2, 6, 24, 135, 930, 7105, 59192, 549360, 5746080, 66713361, 839528052, 11308954657, 163038260294, 2520332282910, 41640324943968, 730119174449151, 13507292654421390, 263004450921933817, 5385277610047242620, 115775314245285797256, 2606072891349667903152, 61248210450060537498321

Offset: 0

Mélika Tebni, Jul 06 2021

nonn

			sqrt(-1+2/(1-x)/exp(x)) =  1 + x^2/2! + 2*x^3/3! + 6*x^4/4! + 24*x^5/5! + 135*x^6/6! + 930*x^7/7! + 7105*x^8/8! + 59192*x^9/9! + ...
a(23) = Sum_{k=1..11} (-1)^(k-1)*A014304(k-1)*A008306(23,k) = 2606072891349667903152.
For k=1, (-1)^(1-1)*A014304(1-1)*A008306(23,1) == -1 (mod 23), because A014304(0) = 1 and A008306(23,1) = (23-1)!
For k>=2, (-1)^(k-1)*A014304(k-1)*A008306(23,k) == 0 (mod 23), because A008306(23,k) == 0 (mod 23), result a(23) == -1 (mod 23).
a(18) = Sum_{k=1..9} (-1)^(k-1)*A014304(k-1)*A008306(18,k) = 730119174449151.
a(18) == 0 (mod (18-1)), because for k >= 1, A008306(18,k) == 0 (mod 17).

Cf. A000166, A008306, A014304, A327006, A345697, A345969, A346119.

A014304:= proc(n) option remember; `if`(n=0, 1, (-1)^n + add(binomial(n,k)*A014304(k)* A014304(n-k-1), k=0..n-1)) end:
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a:= n-> add(((-1)^(k-1)*A014304(k-1)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(sqrt(-1+2/(1-x)/exp(x)), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);

CoefficientList[Series[Sqrt[-1+2/(1-x)/E^x], {x, 0, 24}], x] * Range[0, 24]!

my(x='x+O('x^30)); Vec(serlaplace(sqrt(-1 + 2 / (1 - x) / exp(x)))) \\ Michel Marcus, Jul 06 2021

E.g.f. y(x) satisfies y*y' = exp(-x)*x/(1-x)^2.
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1)*A014304(k-1)*A008306(n,k) for n > 0.
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
a(n) ~ 2 * n^n / exp(n + 1/2). - Vaclav Kotesovec, Jul 06 2021

A358622 Regular triangle read by rows. T(n, k) = [[n, k]], where [[n, k]] are the second order Stirling cycle numbers (or second order reciprocal Stirling numbers). T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 0, 24, 20, 0, 0, 0, 0, 120, 130, 15, 0, 0, 0, 0, 720, 924, 210, 0, 0, 0, 0, 0, 5040, 7308, 2380, 105, 0, 0, 0, 0, 0, 40320, 64224, 26432, 2520, 0, 0, 0, 0, 0, 0, 362880, 623376, 303660, 44100, 945, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Nov 23 2022

[[n, k]] are the number of permutations of an n-set having at least two elements in each orbit. These permutations have no fixed points and therefore [[n, k]] is the number of k-orbit derangements of an n-set. This is the definition and notation (doubling the stacked delimiters of the Stirling cycle numbers) as given by Fekete (see link).
The formal definition expresses the second order Stirling cycle numbers as a binomial sum over second order Eulerian numbers (see the first formula below). The terminology 'associated Stirling numbers of first kind' used elsewhere should be dropped in favor of the more systematic one used here.
Also the Bell transform of the factorial numbers with 0! = 0. For the definition of the Bell transform see A264428.

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     0;
[2] 0,     1,     0;
[3] 0,     2,     0,     0;
[4] 0,     6,     3,     0,    0;
[5] 0,    24,    20,     0,    0,  0;
[6] 0,   120,   130,    15,    0,  0,  0;
[7] 0,   720,   924,   210,    0,  0,  0,  0;
[8] 0,  5040,  7308,  2380,  105,  0,  0,  0,  0;
[9] 0, 40320, 64224, 26432, 2520,  0,  0,  0,  0,  0;

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994, thirty-fourth printing 2022.

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. 8.
Antal E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

A008306 is an irregular subtriangle with more information.
Cf. A000166 (row sums), A024000 (alternating row sums).
Cf. A130534, A201637, A341101, A264428, A269940.

P := (n, x) -> (-x)^n*hypergeom([-n, x], [], 1/x):
row := n -> seq(coeff(simplify(P(n, x)), x, k), k = 0..n):
for n from 0 to 9 do row(n) od;
# Alternative:
T := (n, k) -> add(binomial(n, k - j)*abs(Stirling1(n - k + j, j))*(-1)^(k - j), j =  0..k): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Using the e.g.f.:
egf := (exp(t)*(1 - t))^(-z): ser := series(egf, t, 12):
seq(print(seq(n!*coeff(coeff(ser, t, n), z, k), k=0..n)), n = 0..9);
# Using second order Eulerian numbers:
A358622 := proc(n, k) local j;
add(binomial(j, n - 2*k)*combinat:-eulerian2(n - k, j), j = 0..n-k) end:
seq(seq(A358622(n, k), k = 0..n), n = 0..12);
# Using generalized Laguerre polynomials:
P := (n, x) -> (-1)^n*n!*LaguerreL(n, -n - x, -x):
row := n -> seq(coeff(simplify(P(n, x)), x, k), k = 0..n):
seq(print(row(n)), n = 0..9);

# recursion over rows
from functools import cache
@cache
def StirlingCycleOrd2(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 0]
    rov: list[int] = StirlingCycleOrd2(n - 2)
    row: list[int] = StirlingCycleOrd2(n - 1) + [0]
    for k in range(1, n // 2 + 1):
        row[k] = (n - 1) * (rov[k - 1] + row[k])
    return row
for n in range(9): print(StirlingCycleOrd2(n))
# Alternative, using function BellMatrix from A264428.
from math import factorial
def f(k: int) -> int:
    return factorial(k) if k > 0 else 0
print(BellMatrix(f, 9))

T(n, k) = Sum_{j=0..n-k} binomial(j, n - 2*k)*<>, where <> denote the second order Eulerian numbers (extending Knuth's notation).
T(n, k) = [x^n] (-x)^n * hypergeom([-n, x], [], -1/x).
T(n, k) = n!*[z^k][t^n] (exp(t)*(1 - t))^(-z). (Compare with (exp(t)/(1 - t))^z, which is the e.g.f. of the Sylvester polynomials A341101.)
T(n, k) = [x^k] (-1)^n * n! * L(n, -x - n, -x), where L(n, a, x) is the n-th generalized Laguerre polynomial.
T(n, k) = Sum_{j=0..k} binomial(n, k - j)*[n - k + j, j]*(-1)^(k - j), where [n, k] denotes the (signless) Stirling cycle numbers.
T(n, k) = (n - 1) * (T(n-2, k-1) + T(n-1, k)) with suitable boundary conditions.
T(n + k, k) = A269940(n, k), which might be called the Ward cycle numbers.

cp's OEIS Frontend

A345697 Expansion of the e.g.f. sqrt(1 / (2*exp(x) - 2*x*exp(x) - 1)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A345969 Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x))).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A050211 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A050212 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=4.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A050213 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=5.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A079510 Triangle T(n,k) read by rows; related to number of preorders.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A346119 Expansion of the e.g.f. sqrt(2*x*exp(x) - 2*exp(x) + 3).

Views

Author

Keywords

Examples

Crossrefs

A345697 Expansion of the e.g.f. sqrt(1 / (2exp(x) - 2x*exp(x) - 1)).

A346119 Expansion of the e.g.f. sqrt(2xexp(x) - 2*exp(x) + 3).