A001117 -id:A001117 - cp's OEIS frontend

A000920 Differences of 0: 6!*Stirling2(n,6).

0, 0, 0, 0, 0, 720, 15120, 191520, 1905120, 16435440, 129230640, 953029440, 6711344640, 45674188560, 302899156560, 1969147121760, 12604139926560, 79694820748080, 499018753280880, 3100376804676480, 19141689213218880, 117579844328562000

Offset: 1

N. J. A. Sloane

Number of surjections from an n-element set onto a six-element set, with n >= 6. - Mohamed Bouhamida, Dec 15 2007

Number of rows of n colors using exactly six colors. For n=6, the 720 rows are the 720 permutations of ABCDEF. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A001117, A001118, A000918, A000919, A000770.

Column 6 of A019538.

[6^n-Binomial(6,5)*5^n+Binomial(6,4)*4^n-Binomial(6,3)*3^n+Binomial(6,2)*2^n-Binomial(6,1): n in [1..30]]; // Vincenzo Librandi, May 18 2015

720/(-1+z)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1);

CoefficientList[Series[(720*x^5)/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)),{x,0,30}],x] (* Vincenzo Librandi, Apr 11 2012 *)
k=6; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 6!*stirling(n, 6, 2); \\ Altug Alkan, Sep 25 2018

a(n) = Sum((-1)^i*binomial(6, i)*(6-i)^n, i = 0 .. 5).
a(n) = 6^n-C(6,5)*5^n+C(6,4)*4^n-C(6,3)*3^n+C(6,2)*2^n-C(6,1) with n>=6. - Mohamed Bouhamida, Dec 15 2007
G.f.: 720*x^6/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009]
a(n) = 720*A000770(n). - R. J. Mathar, Apr 30 2015
E.g.f.: (exp(x) - 1)^6. - Geoffrey Critzer, May 17 2015

A000498 Eulerian numbers (Euler's triangle: column k=4 of A008292, column k=3 of A173018).

Original entry on oeis.org

1, 26, 302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450, 198410786, 848090912, 3572085255, 14875399450, 61403313100, 251732291184, 1026509354985, 4168403181210, 16871482830550, 68111623139600, 274419271461131, 1103881308184906, 4434992805213952

Offset: 4

N. J. A. Sloane, Mira Bernstein, and Robert G. Wilson v

There are 2 versions of Euler's triangle:
* A008292 Classic version of Euler's triangle used by Comtet (1974).
* A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).
Euler's triangle rows and columns indexing conventions:
* A008292 The rows and columns of the Eulerian triangle are both indexed starting from 1. (Classic version: used in the classic books by Riordan and Comtet.)
* A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0.(Graham et al.)
Number of permutations of n letters with exactly 3 descents.

			There is one permutation of 4 with exactly 3 descents (4321).
There are 26 permutations of 5 with 3 descents: 15432, 21543, 25431, 31542, 32154, 32541, 35421, 41532, 42153, 42531, 43152, 43215, 43251, 43521, 45321, 51432, 52143, 52431, 53142, 53214, 53241, 53421, 54132, 54213, 54231, 54312. - Neven Juric, Jan 21 2010.

L. Comtet, "Permutations by Number of Rises; Eulerian Numbers." §6.5 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 51 and 240-246, 1974.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
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).

T. D. Noe, Table of n, a(n) for n = 4..200
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
F. N. Castro, O. E. González, and L. A. Medina, The p-adic valuation of Eulerian numbers: trees and Bernoulli numbers, 2014.
E. T. Frankel, A calculus of figurate numbers and finite differences, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]
J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971
Nagatomo Nakamura, Pseudo-Normal Random Number Generation via the Eulerian Numbers, Josai Mathematical Monographs, vol. 8, p 85-95, 2015.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
J. Riordan, Review of Frankel (1950) [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Eulerian Number
Robert G. Wilson v, Letter to N. J. A. Sloane, Apr. 1994
Index entries for linear recurrences with constant coefficients, signature (20,-175,882,-2835,6072,-8777,8458,-5204,1848,-288).

Cf. A008292 (classic version of Euler's triangle used by Comtet (1974).)
Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).)
Cf. A066912.

[(6*4^n -6*(n+1)*3^n +3*n*(n+1)*2^n -(n-1)*n*(n+1))/6: n in [4..50]]; // G. C. Greubel, Oct 23 2017

[EulerianNumber(n,3): n in [4..50]]; // G. C. Greubel, Dec 07 2024

A000498:=proc(n); 4^n-(n+1)*3^n+1/2*(n)*(n+1)*2^n-1/6*(n-1)*(n)*(n+1); end:

LinearRecurrence[{20, -175, 882, -2835, 6072, -8777, 8458, -5204, 1848, -288}, {1, 26, 302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450}, 30] (* Jean-François Alcover, Feb 09 2016 *)
Table[Sum[(-1)^k*Binomial[n+1,k]*(4-k)^n, {k,0,3}], {n,4,50}] (* G. C. Greubel, Oct 23 2017 *)

for(n=4,50, print1((6*4^n -6*(n+1)*3^n +3*n*(n+1)*2^n -(n-1)*n*(n+ 1))/6, ", ")) \\ G. C. Greubel, Oct 23 2017

from sage.combinat.combinat import eulerian_number
print([eulerian_number(n,3) for n in range(4,61)]) # G. C. Greubel, Dec 07 2024

From Mike Zabrocki, Nov 12 2004: (Start)
G.f.: x^4*(1 + 6*x - 43*x^2 + 44*x^3 + 52*x^4 - 72*x^5)/((1-x)^4 * (1-2*x)^3 * (1-3*x)^2 * (1-4*x)).
a(n) = (6*4^n - 6*(n + 1)*3^n + 3*(n)*(n + 1)*2^n - (n - 1)*(n)*(n + 1))/6. (End)
If n>3 is prime, then a(n) == 1 (mod n). A generalization: if a_t(n) denote the number of permutations of n letters with exactly t descents (column t+1 of Euler's triangle A008292), then, for prime n>t, we have a(n) == 1 (mod n). - Vladimir Shevelev, Sep 26 2010
E.g.f.: exp(x)*(exp(3*x) - (1 + 3*x)*exp(2*x) + 2*(x + 2*x^2/2!)*exp(x) - x^2/2! - x^3/3!). - Wolfdieter Lang, Apr 17 2017

More terms from Christian G. Bower, May 12 2000

A056454 Number of palindromes of length n using exactly three different symbols.

Original entry on oeis.org

0, 0, 0, 0, 6, 6, 36, 36, 150, 150, 540, 540, 1806, 1806, 5796, 5796, 18150, 18150, 55980, 55980, 171006, 171006, 519156, 519156, 1569750, 1569750, 4733820, 4733820, 14250606, 14250606, 42850116, 42850116, 128746950, 128746950, 386634060, 386634060, 1160688606

Offset: 1

Marks R. Nester

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-6,6).

Cf. A056449, A001117.

[StirlingSecond((n+1) div 2, 3)*Factorial(3): n in [1..40]]; // Vincenzo Librandi, Sep 26 2018

A056454:= n-> 3!*Stirling2(floor((n+1)/2),3); # (C. Ronaldo)

LinearRecurrence[{1,5,-5,-6,6},{0,0,0,0,6},40] (* Harvey P. Dale, Sep 02 2016 *)
k=3; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 3!*stirling((n+1)\2, 3, 2); \\ Altug Alkan, Sep 25 2018

a(n) = 3! * Stirling2( [(n+1)/2], 3).
G.f.: 6*x^5/((1-x)*(1-2*x^2)*(1-3*x^2)). - Colin Barker, May 05 2012
G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-i*x^2), where k=3 is the number of symbols. - Robert A. Russell, Sep 25 2018

A135456 Number of surjections from an n-element set onto a seven-element set.

Original entry on oeis.org

5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960, 248619571200, 2060056318320, 16540688324160, 129568848121440, 995210916336000, 7524340159588560, 56163512390086080, 414847224363337920

Offset: 7

Mohamed Bouhamida, Dec 15 2007

G. C. Greubel, Table of n, a(n) for n = 7..1000
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Column k=7 of A019538 and A131689.

Cf. A000918, A000919, A000920, A001117, A001118.

LinearRecurrence[{28, -322, 1960, -6769, 13132, -13068, 5040}, {5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960}, 25] (* G. C. Greubel, Oct 14 2016 *)

a(n) = 7^n -C(7,6)*6^n +C(7,5)*5^n -C(7,4)*4^n +C(7,3)*3^n -C(7,2)*2^n +C(7,1) with n>=7.
a(n) = A000771(n) * 7!. - Max Alekseyev, Nov 12 2009
G.f.: -5040*x^7/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^7. - Ilya Gutkovskiy, Jun 19 2018

More terms from Max Alekseyev, Nov 12 2009

A056310 Number of reversible strings with n beads using exactly three different colors.

Original entry on oeis.org

0, 0, 3, 18, 78, 273, 921, 2916, 9150, 28065, 85773, 259848, 785778, 2367813, 7128201, 21427956, 64382550, 193326105, 580372293, 1741847328, 5227116378, 15684323853, 47059266081, 141189861996

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=3, the three rows are ABC, ACB, and BAC, being respectively equivalent to CBA, BCA, and CAB, with which they form chiral pairs. - _Robert A. Russell_, Sep 25 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-6,-24,49,6,-66,36).

Cf. A032120, A005418.
Column 3 of A305621.
Equals (A001117 + A056454) / 2 = A001117 - A305623 = A305623 + A056454.

seq(coeff(series(-3*x^3*(12*x^4-5*x^3-4*x^2+1)/((x-1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)),x,n+1), x, n), n = 1..25); # Muniru A Asiru, Sep 27 2018

k=3; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* Robert A. Russell, Nov 25 2017 *)
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 3, 18, 78, 273, 921}, 40] (* Vincenzo Librandi, Sep 27 2018 *)

a(n) = A032120(n) - 3*A005418(n+1) + 3.

G.f.: -3*x^3*(12*x^4 - 5*x^3 - 4*x^2 + 1)/((x - 1)*(2*x - 1)*(3*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)). [Colin Barker, Jul 07 2012]

A133068 Number of surjections from an n-element set to an eight-element set.

Original entry on oeis.org

40320, 1451520, 30240000, 479001600, 6411968640, 76592355840, 843184742400, 8734434508800, 86355926616960, 823172919528960, 7621934141203200, 68937160460313600, 611692004959217280, 5342844138794426880, 46061530905262118400, 392795626402384128000

Offset: 8

Mohamed Bouhamida, Dec 16 2007

G. C. Greubel, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Column k=8 of A019538 and A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A135456.

[&+[(-1)^(8-k)*Binomial(8, k)*k^n: k in [1..n]]: n in [8..25]]; // Vincenzo Librandi, Oct 21 2017

CoefficientList[Series[40320*x^8/((x - 1)*(2*x - 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(7*x - 1)*(8*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 20 2017 *)
Table[Sum[(-1)^(8 - k)*Binomial[8, k]*k^n, {k, 1, 8}], {n, 8, 20}] (* G. C. Greubel, Oct 21 2017 *)

x='x+O('x^50); Vec(40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1))) \\ G. C. Greubel, Oct 20 2017

a(n) = Sum_{k=1..8} ((-1)^(8-k)*binomial(8,k)*k^n).
a(n) = A049434(n) * 8!. - Max Alekseyev, Nov 13 2009
G.f.: 40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^8. - Ilya Gutkovskiy, Jun 19 2018

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

More terms from Max Alekseyev, Nov 13 2009

A133132 Number of surjections from an n-element set to a ten-element set.

Original entry on oeis.org

3628800, 199584000, 6187104000, 142702560000, 2731586457600, 45950224320000, 703098107712000, 10009442963520000, 134672620008326400, 1732015476199008000, 21473732319740064000, 258323865658578720000

Offset: 10

Mohamed Bouhamida, Dec 16 2007

Vincenzo Librandi, Table of n, a(n) for n = 10..1000
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Column k=10 of A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A049435, A135456.

[10^n-10*9^n+45*8^n-120*7^n+210*6^n-252*5^n+210*4^n-120*3^n+45*2^n-10: n in [10..30]]; // Vincenzo Librandi, Apr 11 2012

With[{nn=30},Drop[CoefficientList[Series[(Exp[x]-1)^10,{x,0,nn}],x] Range[0,nn]!,10]] (* Harvey P. Dale, Sep 01 2016 *)

sum(k=1,10,(-1)^(10-k)*binomial(10,k)*k^n)

a(n) = 10^n-10*9^n+45*8^n-120*7^n+210*6^n-252*5^n+210*4^n-120*3^n+45*2^n-10.
a(n) = A049435(n) * 10!. - Max Alekseyev, Nov 13 2009
G.f.: 3628800*x^10/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x)-1)^10. - Alois P. Heinz, May 17 2016

More terms from Max Alekseyev, Nov 13 2009

Formula corrected by Charles R Greathouse IV, Mar 07 2010

A133360 Number of surjections from an n-element set to a nine-element set.

Original entry on oeis.org

362880, 16329600, 419126400, 8083152000, 130456085760, 1863435974400, 24359586451200, 297846188640000, 3457819037312640, 38528927611574400, 415357755774998400, 4358654246117808000, 44733116259693227520

Offset: 9

Mohamed Bouhamida, Dec 21 2007

Index entries for linear recurrences with constant coefficients, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).

Column k=9 of A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A049447, A135456.

a(n) = Sum_{k=1..9} (-1)^(9-k)*binomial(9,k)*k^n.
a(n) = A049447(n) * 9!. - Max Alekseyev, Nov 12 2009
G.f.: -362880*x^9/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^9. - Ilya Gutkovskiy, Jun 19 2018

More terms from Max Alekseyev, Nov 12 2009

A052761 a(n) = 3!nS(n-1,3), where S denotes the Stirling numbers of second kind.

Original entry on oeis.org

0, 0, 0, 0, 24, 180, 900, 3780, 14448, 52164, 181500, 615780, 2052072, 6749028, 21976500, 71007300, 228009696, 728451972, 2317445100, 7346047140, 23213772120, 73156412196, 229989358500, 721474964100, 2258832312144, 7059480120900, 22026886599900

Offset: 0

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

Matthew House, Table of n, a(n) for n = 0..2079
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 718
Index entries for linear recurrences with constant coefficients, signature (12,-58,144,-193,132,-36).

Cf. A000392, A001117, A052749.

spec := [S,{B=Set(Z,1 <= card),S=Prod(B,B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0},Table[3!*n*StirlingS2[n-1,3],{n,30}]] (* Harvey P. Dale, Feb 07 2015 *)

a(n)={if(n>=1, 3!*n*stirling(n-1, 3, 2), 0)} \\ Andrew Howroyd, Aug 08 2020

E.g.f.: exp(x)^3*x - 3*exp(x)^2*x + 3*x*exp(x) - x.
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-36*n^2 - 66*n - 6*n^3 - 36)*a(n) + (11*n^3 + 55*n^2 + 66*n)*a(n+1) + (-6*n^3 - 24*n^2 - 18*n)*a(n+2) + (n^3 + 3*n^2 + 2*n)*a(n+3)}
For n>=2, a(n) = n*(3^(n-1) - 3*2^(n-1) + 3). - Vaclav Kotesovec, Nov 27 2012
O.g.f.: 12*x^4*(2 - 9*x + 11*x^2 - 3*x^3)/((1 - 3*x)^2*(1 - 2*x)^2*(1 - x)^2). - Matthew House, Feb 16 2017 [Corrected by Georg Fischer, May 19 2019]
From Andrew Howroyd, Aug 08 2020: (Start)
a(n) = n*A001117(n-1) for n > 1.
E.g.f.: x*(exp(x) - 1)^3. (End)

Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001

More terms from Harvey P. Dale, Feb 07 2015

A305623 Number of chiral pairs of rows of n colors with exactly 3 different colors.

Original entry on oeis.org

0, 0, 3, 18, 72, 267, 885, 2880, 9000, 27915, 85233, 259308, 783972, 2366007, 7122405, 21422160, 64364400, 193307955, 580316313, 1741791348, 5226945372, 15684152847, 47058746925, 141189342840, 423593188200, 1270831465995, 3812595048993, 11437991207388, 34314376250772, 102943948309287, 308833455491445, 926503630549920, 2779517334002400, 8338565015656035, 25015720816575273, 75047214375967428

Offset: 1

Robert A. Russell, Jun 06 2018

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

			For a(3) = 3, the chiral pairs are ABC-CBA, ACB-BCA, and BAC-CAB.

Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Third column of A305622.
A056454(n) is number of achiral rows of n colors with exactly 3 different colors.
Cf. A001117, A056310.

k=3; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]

a(n) = 3*(stirling(n,3,2)-stirling(ceil(n/2),3,2)); \\ Altug Alkan, Sep 26 2018

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=3 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A001117(n) - A056454(n)) / 2.
a(n) = A001117(n) - A056310(n) = A056310(n) - A056454(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=3 colors used.
G.f.: 3*x^3*(5*x^2-x-1)/(-36*x^6+30*x^5+24*x^4-25*x^3-x^2+5*x-1). - Simon Plouffe, Jun 20 2018

cp's OEIS Frontend

A000920 Differences of 0: 6!*Stirling2(n,6).

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A000498 Eulerian numbers (Euler's triangle: column k=4 of A008292, column k=3 of A173018).

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Magma

Maple

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SageMath

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A056454 Number of palindromes of length n using exactly three different symbols.

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Magma

Maple

Mathematica

PARI

Formula

A135456 Number of surjections from an n-element set onto a seven-element set.

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Mathematica

Formula

Extensions

A056310 Number of reversible strings with n beads using exactly three different colors.

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Programs

Maple

Mathematica

Formula

A133068 Number of surjections from an n-element set to an eight-element set.

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Magma

Mathematica

PARI

Formula

A052761 a(n) = 3!nS(n-1,3), where S denotes the Stirling numbers of second kind.