A000276 -id:A000276 - cp's OEIS frontend

A377246 a(n) = (n!^2n^(n-1)/4) Sum_{k=4..n} A000276(k) / (n^k * (n-k)!).

0, 0, 0, 108, 25200, 6566400, 2263917600, 1070863718400, 695561049469440, 612326076235776000, 716999439503394432000, 1094463733944478334976000, 2136344904330981293005056000, 5240068882948994816402679398400, 15901807526128013295439617984000000, 58888414506334327924778872791367680000, 262906951354695579633857525111586324480000

Offset: 1

Max Alekseyev, Oct 21 2024

nonn

The formula was listed in A174637 by Vladimir Shevelev. However, it produces a different sequence given here. Apparently, it is also related to permanents of (0,1)-matrices.

V. S. Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian).

Cf. A000276, A174637.

a(n) = (n!^2*n^(n-1)/4) * Sum_{k=4..n} A000276(k) / (n^k * (n-k)!).

For n>=3, a(n) = n! * (((n-1)!/4)*A000276(n) + Sum_{k=2..n-1} (-1)^(n+k+1) * binomial(n,k) * k^(n-k) * a(k)/k!).

A008306 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).

Original entry on oeis.org

1, 2, 6, 3, 24, 20, 120, 130, 15, 720, 924, 210, 5040, 7308, 2380, 105, 40320, 64224, 26432, 2520, 362880, 623376, 303660, 44100, 945, 3628800, 6636960, 3678840, 705320, 34650, 39916800, 76998240, 47324376, 11098780, 866250, 10395

Offset: 2

N. J. A. Sloane

Also, T(n,k) is the number of derangements (permutations with no fixed points) of {1..n} with k cycles.

The sum of the n-th row is the n-th subfactorial: A000166(n). - Gary Detlefs, Jul 14 2010

			Rows 2 through 7 are:
    1;
    2;
    6,   3;
   24,  20;
  120, 130,  15;
  720, 924, 210;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.

Reinhard Zumkeller, Rows n = 2..125 of table, flattened
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
W. Carlitz, On some polynomials of Tricomi, Bollettino dell'Unione Matematica Italiana, Serie 3, Vol. 13, (1958), n. 1, p. 58-64
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
W. Gautschi, The incomplete gamma functions since Tricomi (Cf. p. 206-207.)
P. Gniewek and B. Jeziorski, Convergence properties of the multipole expansion of the exchange contribution to the interaction energy, arXiv preprint arXiv:1601.03923 [physics.chem-ph], 2016.
S. Karlin and J. McGregor, Many server queuing processes with Poisson input and exponential service times, Pacific Journal of Mathematics, Vol. 8, No. 1, p. 87-118, March (1958). Cf. p. 117.
R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239. (See 333. From Tom Copeland, Jan 03 2016)
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
N. Temme, A class of polynomials related to those of Laguerre
N. Temme, Traces to Tricomi in recent work on special functions and asymptotics of integrals
A. Topuzoglu, The Carlitz rank of permutations of finite fields: A survey, Journal of Symbolic Computation, Online, Dec 07, 2013.
Eric Weisstein's World of Mathematics, Permutation Cycle
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).

Cf. A000166, A106828 (another version), A079510 (rearranged triangle), A235706 (specializations).
Diagonals: A000142, A000276, A000483.
Diagonals give reversed rows of A111999.

a008306 n k = a008306_tabf !! (n-2) !! (k-1)
a008306_row n = a008306_tabf !! (n-2)
a008306_tabf = map (fst . fst) $ iterate f (([1], [2]), 3) where
   f ((us, vs), x) =
     ((vs, map (* x) $ zipWith (+) ([0] ++ us) (vs ++ [0])), x + 1)
-- Reinhard Zumkeller, Aug 05 2013

A008306 := proc(n,k) local j;
add(binomial(j,n-2*k)*A008517(n-k,j),j=0..n-k) end;
seq(print(seq(A008306(n,k),k=1..iquo(n,2))),n=2..12):
# Peter Luschny, Apr 20 2011

t[0, 0] = 1; t[n_, 0] = 0; t[n_, k_] /; k > n/2 = 0; t[n_, k_] := t[n, k] = (n - 1)*(t[n - 1, k] + t[n - 2, k - 1]); A008306 = Flatten[ Table[ t[n, k], {n, 2, 12}, {k, 1, Quotient[n, 2]}]] (* Jean-François Alcover, Jan 25 2012, after David Callan *)

{ A008306(n,k) = (-1)^(n+k) * sum(i=0,k, (-1)^i * binomial(n,i) * stirling(n-i,k-i,1) ); } \\ Max Alekseyev, Sep 08 2018

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(n,i) * |stirling1(n-i,k-i)| = (-1)^(n+k) * Sum_{i=0..k} (-1)^i * binomial(n,i) * A008275(n-i,k-i). - Max Alekseyev, Sep 08 2018
E.g.f.: 1 + Sum_{1 <= 2*k <= n} T(n, k)*t^n*u^k/n! = exp(-t*u)*(1-t)^(-u).
Recurrence: T(n, k) = (n-1)*(T(n-1, k) + T(n-2, k-1)) for 1 <= k <= n/2 with boundary conditions T(0,0) = 1, T(n,0) = 0 for n >= 1, and T(n,k) = 0 for k > n/2. - David Callan, May 16 2005
E.g.f. for column k: B(A(x)) where A(x) = log(1/1-x)-x and B(x) = x^k/k!.
From Tom Copeland, Jan 05 2016: (Start)
The row polynomials of this signed array are the orthogonal NL(n,x;x-n) = n! Sum_{k=0..n} binomial(x,n-k)*(-x)^k/k!, the normalized Laguerre polynomials of order (x-n) as discussed in Gautschi (the Temme, Carlitz, and Karlin and McGregor references come from this paper) in regard to asymptotic expansions of the upper incomplete gamma function--Tricomi's Cinderella of special functions.
e^(x*t)*(1-t)^x = Sum_{n>=0} NL(n,x;x-n)*x^n/n!.
The first few are
NL(0,x) = 1
NL(1,x) = 0
NL(2,x) = -x
NL(3,x) = 2*x
NL(4,x) = -6*x + 3*x^2.
With D=d/dx, :xD:^n = x^n D^n, :Dx:^n = D^n x^n, and K(a,b,c), the Kummer confluent hypergeometric function, NL(n,x;y-n) = n!*e^x binomial(xD+y,n)*e^(-x) = n!*e^x Sum_{k=0..n} binomial(k+y,n) (-x)^k/k! = e^x x^(-y+n) D^n (x^y e^(-x)) = e^x x^(-y+n) :Dx:^n x^(y-n)*e^(-x) = e^x*x^(-y+n)*n!*L(n,:xD:,0)*x^(y-n)*e^(-x) = n! binomial(y,n)*K(-n,y-n+1,x) = n!*e^x*(-1)^n*binomial(-xD-y+n-1,n)*e^(-x). Evaluate these expressions at y=x after the derivative operations to obtain NL(n,x;x-n). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001

A136394 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 20, 3, 1, 84, 35, 1, 409, 295, 15, 1, 2365, 2359, 315, 1, 16064, 19670, 4480, 105, 1, 125664, 177078, 56672, 3465, 1, 1112073, 1738326, 703430, 74025, 945, 1, 10976173, 18607446, 8941790, 1346345, 45045, 1, 119481284, 216400569, 118685336

Offset: 0

Vladeta Jovovic, May 03 2008

			Triangle (n,k) begins:
  1;
  1;
  1,    1;
  1,    5;
  1,   20,    3;
  1,   84,   35;
  1,  409,  295,  15;
  1, 2365, 2359, 315;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021. See Theorem 2. p. 5.
FindStat - Combinatorial Statistic Finder, The number of nontrivial cycles of a permutation pi in its cycle decomposition
Bin Han, Jianxi Mao, and Jiang Zeng, Equidistributions around special kinds of descents and excedances, arXiv:2103.13092 [math.CO], 2021, see page 2.

Columns k=0-10 give: A000012, A006231, A289950, A289951, A289952, A289953, A289954, A289955, A289956, A289957, A289958.
Row sums give A000142.
Cf. A000276, A000483, A001147, A001705, A051577, A124324, A159964.

egf:= proc(k::nonnegint) option remember; x-> exp(x)* ((-x-ln(1-x))^k)/k! end; T:= (n,k)-> coeff(series(egf(k)(x), x=0, n+1), x, n) *n!; seq(seq(T(n,k), k=0..n/2), n=0..30); # Alois P. Heinz, Aug 14 2008
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
      `if`(i>1, x, 1)*binomial(n-1, i-1)*(i-1)!, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 25 2016
# third Maple program:
T:= proc(n, k) option remember; `if`(k<0 or k>2*n, 0,
      `if`(n=0, 1, add(T(n-i, k-`if`(i>1, 1, 0))*
       mul(n-j, j=1..i-1), i=1..n)))
    end:
seq(seq(T(n,k), k=0..n/2), n=0..15);  # Alois P. Heinz, Jul 16 2017

max = 12; egf = Exp[x*(1-y)]/(1-x)^y; s = Series[egf, {x, 0, max}, {y, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]*n!; t[0, 0] = t[1, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)

E.g.f.: exp(x*(1-y))/(1-x)^y. Binomial transform of triangle A008306. exp(x)*((-x-log(1-x))^k)/k! is e.g.f. of k-th column.
From Alois P. Heinz, Jul 13 2017: (Start)
T(2n,n) = A001147(n).
T(2n+1,n) = A051577(n) = (2*n+3)!!/3 = A001147(n+2)/3. (End)
From Alois P. Heinz, Aug 17 2023: (Start)
Sum_{k=0..floor(n/2)} k * T(n,k) = A001705(n-1) for n>=1.
Sum_{k=0..floor(n/2)} (-1)^k * T(n,k) = A159964(n-1) for n>=1. (End)

A000483 Associated Stirling numbers: second-order reciprocal Stirling numbers (Fekete) a(n) = [[n, 3]]. The number of 3-orbit permutations of an n-set with at least 2 elements in each orbit.

Original entry on oeis.org

15, 210, 2380, 26432, 303660, 3678840, 47324376, 647536032, 9418945536, 145410580224, 2377609752960, 41082721413120, 748459539843840, 14345340443665920, 288650580508961280, 6085390148673177600, 134167064248901376000, 3088040233895705088000, 74077507611407752704000, 1849221425299053367296000

Offset: 6

N. J. A. Sloane

nonn

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).

Alois P. Heinz, Table of n, a(n) for n = 6..200
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

Cf. A000907, A001784, A001785. A diagonal of triangle in A008306.

Cf. A000276.

nn=25;a=Log[1/(1-x)]-x;Drop[Range[0,nn]!CoefficientList[Series[a^3/3!, {x,0,nn}],x],6]  (* Geoffrey Critzer, Nov 03 2012 *)

With alternating signs: Ramanujan polynomials psi_4(n-3, x) evaluated at 1. - Ralf Stephan, Apr 16 2004
E.g.f.: -((x+log(1-x))^3)/6. - Vladeta Jovovic, May 03 2008
Conjecture: (n-2)*(n-4)*a(n) -(n-1)*(3*n^2-21*n+35)*a(n-1) +(n-1)*(n-2)*(3*n^2-24*n+47)*a(n-2) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 18 2015
Conjecture: 3*(-n+4)*a(n) +(9*n^2-59*n+90)*a(n-1) +(-9*n^3+96*n^2-348*n+436)*a(n-2) +(n-3)*(3*n^3-45*n^2+237*n-430)*a(n-3) +5*(n-5)*(n-6)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 18 2015

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

More terms from Sean A. Irvine, Nov 14 2010

A259456 Triangle read by rows, giving coefficients in an expansion of absolute values of Stirling numbers of the first kind in terms of binomial coefficients.

Original entry on oeis.org

1, 2, 3, 6, 20, 15, 24, 130, 210, 105, 120, 924, 2380, 2520, 945, 720, 7308, 26432, 44100, 34650, 10395, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025, 362880, 6636960, 47324376, 177331440, 389449060, 520059540, 416215800

Offset: 0

N. J. A. Sloane, Jun 30 2015

			Triangle begins:
1,
2,3,
6,20,15,
24,130,210,105,
120,924,2380,2520,945,
...
For k=4 and j=2 in Knuth's equation, |S1(4,4-2)| = |S1(4,2)| = |A008275(4,2)| = 11 = p_{2,1}*C(4,3) +p_{2,2}*C(4,4) = 2*4+3*1. - _R. J. Mathar_, Jul 16 2015

L. Comtet, Advanced Combinatorics (1974), Chapter VI, page 256.
DJ Jeffrey, GA Kalugin, N Murdoch, Lagrange inversion and Lambert W, Preprint 2015; http://www.apmaths.uwo.ca/~djeffrey/Offprints/JeffreySYNASC2015paper17.pdf
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}.

Lothar Berg, On polynomials related with generalized Bernoulli numbers, Rostock Math. Kolloq. (2002).
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010) #10.4.4 page 4.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
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.
Donald E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
Donald E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992.
Richard B. Paris, An asymptotic approximation for incomplete Gaussian sums. II., J. Comp. Appl. Math 212 (2008) 16-30, Table 1.
Grzegorz Rzadkowski, On some expansions for the Euler Gamma function and the Riemann Zeta function, arxiv:1007.1955 [math.CA], Table 1. J. Comp. Appl. Math. 236 (15) (2012), 3710-3719.
Lajos Takács, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581. Table 3 gives a version of the triangle.

Cf. This is a row reversed and unsigned version of A111999.
Cf. A008275, A000276 (2nd column), A000483 (3rd column), A000142 (1st column).
Cf. A133932.

A259456 := proc(n,k)
    option remember;
    if k < 1 or k > n  then
        0 ;
    elif n = 1 then
        1;
    else
        procname(n-1,k-1)+procname(n-1,k);
        %*(n+k-1) ;
    end if;
end proc:
seq(seq(A259456(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Jul 18 2015

T[n_, k_] := T[n, k] = If[k < 1 || k > n, 0, If[n == 1, 1, (T[n-1, k-1] + T[n-1, k])(n+k-1)]];
Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *)

T(n,k) = (n-k-1)*( T(n-1,k-1)+T(n-1,k) ), n>=1, 1<=k<=n. [Berg, Eq. 6]

The general results on the convolution of the refined partition polynomials of A133932, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these unsigned polynomials. - Tom Copeland, Sep 20 2016

A174637 Number of n X n (0,1) matrices with two 1's in each row the permanent of which equals to 4.

Original entry on oeis.org

0, 0, 0, 18, 2400, 325800, 52496640, 10304300160, 2458401684480, 705918026419200, 241147866161664000, 96890287539173990400, 45304089884519168102400, 24415719893124157985587200, 15035096121857624246353920000, 10496828397482345253454479360000, 8250414679239607850470753370112000

Offset: 1

Vladimir Shevelev, Mar 25 2010

nonn

If a (0,1) matrix with two 1's in each row has positive permanent, then it equals to a power of 2.

V. S. Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian).

Max Alekseyev, Table of n, a(n) for n = 1..100

Cf. A000276, A001866, A174638, A174586, A377246.

a174637(n) = n!*(n-1)!/4 * sum(l=0,n-4, n^l/l! * sum(i=2, n-l-2, 1/i)); \\ Max Alekseyev, Oct 21 2024

a(n) = n!/4 * Sum_{l=0..n-4} binomial(n-1,l) * n^l * A000276(n-l). - Max Alekseyev, Oct 21 2024

G.f. for 4*a(n)/n!/(n-1)!: (W(-x)-ln(1+W(-x)))*(W(-x)/(1+W(-x)))^2, where W() is Lambert W-function. - Max Alekseyev, Oct 21 2024

Incorrect formula moved to A377246 and terms a(15) onward added by Max Alekseyev, Oct 21 2024

A052518 Number of pairs of cycles of cardinality at least 2.

Original entry on oeis.org

0, 0, 0, 0, 6, 40, 260, 1848, 14616, 128448, 1246752, 13273920, 153996480, 1935048960, 26193473280, 380120670720, 5888620684800, 97007636275200, 1693590745190400, 31237853849395200, 607035345406156800

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..445
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 84

Cf. A000254, A000276.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Log(1-x)^2 + 2*x*Log(1-x) + x^2 )); [0,0,0,0] cat [Factorial(n+3)*b[n]: n in [1..m-4]]; // G. C. Greubel, May 13 2019

Pairs spec := [S,{B=Cycle(Z,2 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{m = 25}, CoefficientList[Series[Log[1-x]^2 +2*x*Log[1-x] +x^2, {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 13 2019 *)

a(n) = if (n <= 2, 0, round(2*(n-2)!*((n-1)*(psi(n)+Euler)-n))); \\ Michel Marcus, Jul 08 2015

my(x='x+O('x^25)); concat(vector(4), Vec(serlaplace( log(1-x)^2 + 2*x*log(1-x) + x^2 ))) \\ G. C. Greubel, May 13 2019

m = 25; T = taylor(log(1-x)^2 + 2*x*log(1-x) + x^2, x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

E.g.f.: log(1-x)^2 + 2*x*log(1-x) + x^2.
n*a(n+2) + (1-n-2*n^2)*a(n+1) - n*(1-n^2)*a(n) = 0, with a(0) = ... = a(3) = 0, a(4) = 3!.
a(n) = 2*(n-2)!*((n-1)*(Psi(n) + gamma) - n), n>2. - Vladeta Jovovic, Sep 21 2003

cp's OEIS Frontend

A377246 a(n) = (n!^2*n^(n-1)/4) * Sum_{k=4..n} A000276(k) / (n^k * (n-k)!).

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A008306 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).

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A136394 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

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A000483 Associated Stirling numbers: second-order reciprocal Stirling numbers (Fekete) a(n) = [[n, 3]]. The number of 3-orbit permutations of an n-set with at least 2 elements in each orbit.

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A259456 Triangle read by rows, giving coefficients in an expansion of absolute values of Stirling numbers of the first kind in terms of binomial coefficients.

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Keywords

Examples

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Links

Crossrefs

Programs

Maple

Mathematica

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A174637 Number of n X n (0,1) matrices with two 1's in each row the permanent of which equals to 4.

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PARI

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A052518 Number of pairs of cycles of cardinality at least 2.

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Magma

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A377246 a(n) = (n!^2n^(n-1)/4) Sum_{k=4..n} A000276(k) / (n^k * (n-k)!).