A000392 -id:A000392 - cp's OEIS frontend

A016218 Expansion of 1/((1-x)(1-4x)(1-5x)).

1, 10, 71, 440, 2541, 14070, 75811, 400900, 2091881, 10808930, 55442751, 282806160, 1436400421, 7271480590, 36715316891, 185008240220, 930767824161, 4676745613050, 23475354034231, 117743274047080, 590182385739101, 2956775990710310, 14807336201610771

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-29,20).

Cf. A016208, A000392, A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A016256.

Vec(1/((1-x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

From Vincenzo Librandi, Feb 10 2011: (Start)
a(n) = a(n-1) + 5^(n+1) - 4^(n+1), n >= 1.
a(n) = 9*a(n-1) - 20*a(n-2) + 1, n >= 2. (End)
a(n) = 1/12 - 4^(n+2)/3 + 5^(n+2)/4. - R. J. Mathar, Mar 15 2011

A056327 Number of reversible string structures with n beads using exactly three different colors.

Original entry on oeis.org

0, 0, 1, 4, 15, 50, 160, 502, 1545, 4730, 14356, 43474, 131145, 395150, 1188580, 3572902, 10732065, 32225810, 96733636, 290322394, 871200825, 2614097750, 7843255300, 23531775502, 70599259185, 211805902490

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly three different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

			For a(4)=4, the color patterns are ABCA, ABBC, AABC, and ABAC. The first two are achiral. - _Robert A. Russell_, Oct 14 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]

G. C. Greubel, 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).

Column 3 of A284949.
Cf. A056310.
Cf. A000392 (oriented), A320526 (chiral), A304973 (achiral).

I:=[0,0,1,4,15,50,160]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -24*Self(n-3) +49*Self(n-4) +6*Self(n-5) -66*Self(n-6) +36*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018

k=3; Table[(StirlingS2[n,k] + If[EvenQ[n], 2StirlingS2[n/2+1,3] - 2StirlingS2[n/2,3], StirlingS2[(n+3)/2,3] - StirlingS2[(n+1)/2,3]])/2, {n,30}] (* Robert A. Russell, Oct 15 2018 *)
Ach[n_, k_] := Ach[n, k] = If[n < 2, Boole[n == k && n >= 0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]
k=3; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n,30}] (* Robert A. Russell, Oct 15 2018 *)
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 1, 4, 15, 50, 160}, 30] (* Robert A. Russell, Oct 15 2018 *)

m=40; v=concat([0,0,1,4,15,50,160], vector(m-7)); for(n=8, m, v[n] = 6*v[n-1] -6*v[n-2] -24*v[n-3] +49*v[n-4] +6*v[n-5] -66*v[n-6] +36*v[n-7] ); v \\ G. C. Greubel, Oct 16 2018

a(n) = A001998(n-1) - A005418(n).
G.f.: x^3*(3*x^4 - 8*x^3 + 3*x^2 + 2*x - 1)/((x-1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)). - Colin Barker, Sep 23 2012
From Robert A. Russell, Oct 14 2018: (Start)
a(n) = (S2(n,k) + A(n,k))/2, where k=3 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
a(n) = (A000392(n) + A304973(n)) / 2 = A000392(n) - A320526(n) = A320526(n) + A304973(n). (End)

A134169 a(n) = 2^(n-1)*(2^n - 1) + 1.

Original entry on oeis.org

1, 2, 7, 29, 121, 497, 2017, 8129, 32641, 130817, 523777, 2096129, 8386561, 33550337, 134209537, 536854529, 2147450881, 8589869057, 34359607297, 137438691329, 549755289601, 2199022206977, 8796090925057, 35184367894529

Offset: 0

Ross La Haye, Jan 12 2008

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either (Case 0) x and y are disjoint, x is not a subset of y, and y is not a subset of x; or (Case 1) x and y are intersecting, but x is not a subset of y, and y is not a subset of x; or (Case 2) x and y are intersecting, and either x is a proper subset of y, or y is a proper subset of x; or (Case 3) x = y.

			a(2) = 7 because for P(A) = {{},{1},{2},{1,2}} we have for Case 0 {{1},{2}}; we have for Case 2 {{1},{1,2}}, {{2},{1,2}}; and we have for Case 3 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under Case 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A000392, A032263, A028243, A000079, A006516.

Table[EulerE[2,2^n],{n,0,60}]/2+1 (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
LinearRecurrence[{7,-14,8},{1,2,7},30] (* Harvey P. Dale, Mar 12 2013 *)

a(n) = 2^(n-1)*(2^n - 1) + 1.
a(n) = StirlingS2(2^n,2^n - 1) + 1 = C(2^n,2) + 1 = A006516(n) + 1.
From R. J. Mathar, Feb 15 2010: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: (1 - 5*x + 7*x^2)/((1-x) * (2*x-1) * (4*x-1)). (End)

More terms from Vladimir Joseph Stephan Orlovsky, Nov 03 2009

Edited by N. J. A. Sloane, Jan 25 2015 at the suggestion of Michel Marcus

A218577 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with maximal element k-1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 11, 1, 1, 31, 90, 74, 20, 1, 1, 63, 301, 402, 209, 37, 1, 1, 127, 966, 1951, 1629, 590, 70, 1, 1, 255, 3025, 8869, 10839, 6430, 1685, 135, 1, 1, 511, 9330, 38720, 65720, 56878, 25313, 4870, 264, 1

Offset: 1

Joerg Arndt, Nov 03 2012

Row sums are A022493.
Second column is A000225 (2^n - 1).
Third column appears to be A000392 (Stirling numbers S(n,3)).
Second diagonal (from the right) appears to be A006127 (2^n + n).

			Triangle starts:
1;
1,    1;
1,    3,     1;
1,    7,     6,      1;
1,   15,    25,     11,      1;
1,   31,    90,     74,     20,      1;
1,   63,   301,    402,    209,     37,      1;
1,  127,   966,   1951,   1629,    590,     70,     1;
1,  255,  3025,   8869,  10839,   6430,   1685,   135,     1;
1,  511,  9330,  38720,  65720,  56878,  25313,  4870,   264,   1;
1, 1023, 28501, 164676, 376114, 444337, 292695, 99996, 14209, 521, 1;
...
The 53 ascent sequences of length 5 are (dots for zeros):
[ #]     ascent-seq.   #max digit
[ 1]    [ . . . . . ]   0
[ 2]    [ . . . . 1 ]   1
[ 3]    [ . . . 1 . ]   1
[ 4]    [ . . . 1 1 ]   1
[ 5]    [ . . . 1 2 ]   2
[ 6]    [ . . 1 . . ]   1
[ 7]    [ . . 1 . 1 ]   1
[ 8]    [ . . 1 . 2 ]   2
[ 9]    [ . . 1 1 . ]   1
[10]    [ . . 1 1 1 ]   1
[11]    [ . . 1 1 2 ]   2
[12]    [ . . 1 2 . ]   2
[13]    [ . . 1 2 1 ]   2
[14]    [ . . 1 2 2 ]   2
[15]    [ . . 1 2 3 ]   3
[16]    [ . 1 . . . ]   1
[17]    [ . 1 . . 1 ]   1
[18]    [ . 1 . . 2 ]   2
[19]    [ . 1 . 1 . ]   1
[20]    [ . 1 . 1 1 ]   1
[21]    [ . 1 . 1 2 ]   2
[22]    [ . 1 . 1 3 ]   3
[23]    [ . 1 . 2 . ]   2
[24]    [ . 1 . 2 1 ]   2
[25]    [ . 1 . 2 2 ]   2
[26]    [ . 1 . 2 3 ]   3
[27]    [ . 1 1 . . ]   1
[28]    [ . 1 1 . 1 ]   1
[29]    [ . 1 1 . 2 ]   2
[...]
[49]    [ . 1 2 3 . ]   3
[50]    [ . 1 2 3 1 ]   3
[51]    [ . 1 2 3 2 ]   3
[52]    [ . 1 2 3 3 ]   3
[53]    [ . 1 2 3 4 ]   4
There is 1 sequence with maximum zero, 15 with maximum one, etc.,
therefore the fifth row is 1, 15, 25, 11, 1.

Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened (first 15 rows from Joerg Arndt)
Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, Sergey Kitaev, (2+2)-free posets, ascent sequences and pattern avoiding permutations, arXiv:0806.0666 [math.CO], 2008-2009.
William Y. C. Chen, Alvin Y.L. Dai, Theodore Dokos, Tim Dwyer and Bruce E. Sagan, On 021-Avoiding Ascent Sequences, The Electronic Journal of Combinatorics Volume 20, Issue 1 (2013), #P76.

Cf. A022493 (number of ascent sequences), A137251 (ascent sequences with k ascents), A175579 (ascent sequences with k zeros).

Cf. A218579 (ascent sequences with last zero at position k-1), A218580 (ascent sequences with first occurrence of the maximal value at position k-1), A218581 (ascent sequences with last occurrence of the maximal value at position k-1).

A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

Original entry on oeis.org

1, 3, 10, 33, 106, 333, 1030, 3153, 9586, 29013, 87550, 263673, 793066, 2383293, 7158070, 21490593, 64504546, 193579173, 580868590, 1742867913, 5229128026, 15688432653, 47067395110, 141206379633, 423627527506, 1270899359733

Offset: 1

Vladeta Jovovic and Goran Kilibarda, Feb 28 2000

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 3) x = y. - Ross La Haye, Jan 12 2008
From Paul Barry, Apr 27 2003: (Start)
With offset 0, this is a(n) = (3*3^n - 2*2^n + 1)/2.
G.f. (1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f. (3*exp(3*x) - 2*exp(2*x) + exp(x))/2.
Binomial transform of A083329.
Second binomial transform of A040001. (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000225, A000392, A028243, A000079.
Cf. A036239.
Column k=2 of A288638.
Third column of A294201.

[(3^n-2^n+1)/2: n in [1..30]]; // G. C. Greubel, Oct 06 2017

A053156:=n->(3^n - 2^n + 1)/2: seq(A053156(n), n=1..40); # Wesley Ivan Hurt, Oct 06 2017

LinearRecurrence[{6,-11,6}, {1, 3, 10}, 50] (* or *) Table[(3^n - 2^n + 1)/2, {n,1,50}] (* G. C. Greubel, Oct 06 2017 *)

a(n) = (3^n-2^n+1)/2; \\ Michel Marcus, Nov 30 2015

a(n) = (3^n - 2^n + 1)/2.
a(n) = StirlingS2(n+2,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 12 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
G.f.: x*(1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)

A062254 3rd level triangle related to Eulerian numbers and binomial transforms (A062253 is second level, triangle of Eulerian numbers is first level and triangle with Z(0,0)=1 and Z(n,k)=0 otherwise is 0th level).

Original entry on oeis.org

1, 6, 0, 25, 10, 0, 90, 120, 15, 0, 301, 896, 406, 21, 0, 966, 5376, 5586, 1176, 28, 0, 3025, 28470, 55560, 27910, 3123, 36, 0, 9330, 139320, 456525, 437100, 122520, 7860, 45, 0, 28501, 646492, 3312078, 5339719, 2912833, 494802, 19096, 55, 0

Offset: 0

Henry Bottomley, Jun 14 2001

Binomial transform of n^3*k^n is ((kn)^3 + 3(kn)^2 + (1 - k)(kn))*(k + 1)^(n - 3); of n^4*k^n is ((kn)^4 + 6(kn)^3 + (7 - 4k)(kn)^2 + (1 - 4k + k^2)(kn))*(k + 1)^(n - 4); of n^5*k^n is ((kn)^5 + 10(kn)^4 + (25 - 10k)(kn)^3 + (15 - 30k + 5k^2)(kn)^2 + (1 - 11k + 11k^2 - k^3)(kn))*(k + 1)^(n - 5); of n^6*k^n is ((kn)^6 + 15(kn)^5 + (65 - 20k)(kn)^4 + (90 - 120k + 15k^2)(kn)^3 + (31 - 146k + 91k^2 - 6k^3)(kn)^2 + (1 - 26k + 66k^2 - 26k^3 + k^4)(kn))*(k + 1)^(n - 6). This sequence gives the (unsigned) polynomial coefficients of (kn)^3.

			Rows start:
 (1),
 (6,0),
 (25,10,0),
 (90,120,15,0),
 ...

First column is A000392. Diagonals include A000007 and all but the start of A000217. Row sums are A000399.
Taking all the levels together to create a pyramid, one face would be A010054 as a triangle with a parallel face which is Pascal's triangle (A007318) with two columns removed, another face would be a triangle of Stirling numbers of the second kind (A008277) and a third face would be A000007 as a triangle, (cont.)
(cont.) with a triangle of Eulerian numbers (A008292), A062253, A062254 and A062255 as faces parallel to it. The row sums of this last group would provide a triangle of unsigned Stirling numbers of the first kind (A008275).

E(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, (k+1)*E(n-1, k)+(n-k)*E(n-1, k-1)));
A2(n, k) = if ((n<0) || (k<0), 0, (k+2)*A2(n-1, k)+(n-k)*A2(n-1, k-1)+E(n, k));
A3(n, k) = if ((n<0) || (k<0), 0, (k+3)*A3(n-1, k)+(n-k)*A3(n-1, k-1) + A2(n, k));
row3(n) = vector(n+1, k, A3(n,k-1)); \\ Michel Marcus, Jan 27 2025

A(n, k) = (k+3)*A(n-1, k) + (n-k)*A(n-1, k-1) + A062253(n, k).

A094374 a(n) = (3^n-1)/2 + 2^n.

Original entry on oeis.org

1, 3, 8, 21, 56, 153, 428, 1221, 3536, 10353, 30548, 90621, 269816, 805353, 2407868, 7207221, 21588896, 64701153, 193972388, 581655021, 1744440776, 5232273753, 15694724108, 47079978021, 141231545456, 423677859153, 1271000023028, 3812932960221, 11438664662936

Offset: 0

Paul Barry, Apr 28 2004

Binomial transform of A094373.
Row sums of A125103. - Paul Barry, Dec 04 2007
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x = y. - Ross La Haye, Jan 11 2008
a(n) is the number of words of length n over the alphabet {0,1,2} with an even number of occurrences of the substring 01. - Daimon S. Mayorga, Sep 10 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
A. Prasad, Equivalence classes of nodes in trees and rational generating functions, arXiv preprint arXiv:1407.5284 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000079, A000225, A000392, A003462, A094373, A125103.

[(3^n-1)/2+2^n: n in [0..30]]; // Vincenzo Librandi, Nov 30 2015

Table[(3^n-1)/2+2^n,{n,0,30}] (* or *) LinearRecurrence[{6,-11,6},{1,3,8},30] (* Harvey P. Dale, Jul 22 2013 *)

a(n)=(3^n-1)/2+2^n \\ Charles R Greathouse IV, Oct 16 2015

[(3^n +2^(n+1) -1)//2 for n in range(31)] # G. C. Greubel, Sep 26 2024

G.f.: (1-3x+x^2)/((1-x)*(1-2x)*(1-3x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
a(n) = A003462(n) + A000079(n).
a(n) = Sum_{k=0..n} C(n,k)+2^k*C(n,k+1). - Paul Barry, Dec 04 2007
a(n) = StirlingS2(n+1,3) + 2*StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 11 2008
E.g.f.: exp(2*x)*(1 + sinh(x)). - G. C. Greubel, Sep 26 2024

A106800 Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 1, 0, 1, 28, 266, 1050, 1701, 966, 127, 1, 0, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 0, 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1, 0

Offset: 0

N. J. A. Sloane

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 19 2005

			From _Gheorghe Coserea_, Jan 30 2017: (Start)
Triangle starts:
  n\k  [0]  [1]   [2]    [3]    [4]    [5]    [6]   [7] [8] [9]
  [0]   1;
  [1]   1,   0;
  [2]   1,   1,    0;
  [3]   1,   3,    1,     0;
  [4]   1,   6,    7,     1,     0;
  [5]   1,  10,   25,    15,     1,     0;
  [6]   1,  15,   65,    90,    31,     1,     0;
  [7]   1,  21,  140,   350,   301,    63,     1,    0;
  [8]   1,  28,  266,  1050,  1701,   966,   127,    1,  0;
  [9]   1,  36,  462,  2646,  6951,  7770,  3025,  255,  1,  0;
  ...
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, table 2.14.1 at page 24.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 2014.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
Eric Weisstein's World of Mathematics, Bell Polynomial.

See A008277 and A048993, which are the main entries for this triangle of numbers.
The Stirling1 counterpart is A054654.
Row sum: A000110.
Column 0: A000012.
Column 1: A000217.
Main Diagonal: A000007.
1st minor diagonal: A000012.
2nd minor diagonal: A000225.
3rd minor diagonal: A000392.
Cf. A008278, A340556.

seq(seq(Stirling2(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

Table[ StirlingS2[n, m], {n, 0, 10}, {m, n, 0, -1}]//Flatten (* Robert G. Wilson v, Jan 30 2017 *)

N=11; x='x+O('x^N); t='t; concat(apply(p->Vec(p), Vec(serlaplace(exp(t*(exp(x)-1))))))  \\ Gheorghe Coserea, Jan 30 2017
{T(n, k) = my(A, B); if( n<0 || k>n, 0, A = B = exp(x + x * O(x^n)); for(i=1, n, A = x * A'); polcoeff(A / B, n-k))}; /* Michael Somos, Aug 16 2017 */

flatten([[stirling_number2(n, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 11 2021

A(x;t) = exp(t*(exp(x)-1)) = Sum_{n>=0} P_n(t) * x^n/n!, where P_n(t) = Sum_{k=0..n} T(n,k)*t^(n-k). - Gheorghe Coserea, Jan 30 2017
Also, P_n(t) * exp(t) = (t * d/dt)^n exp(t). - Michael Somos, Aug 16 2017
T(n, k) = Sum_{j=0..k} E2(k, j)*binomial(n + k - j, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021

A320526 a(n) is the number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 3 colors (subsets).

Original entry on oeis.org

0, 0, 0, 2, 10, 40, 141, 464, 1480, 4600, 14145, 43052, 130480, 393820, 1186521, 3568784, 10725760, 32213200, 96714465, 290284052, 871142800, 2613981700, 7843080201, 23531425304, 70598731840, 211804847800, 635432109585, 1906330676252, 5719061512720, 17157321139180

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(4)=2, the two chiral pairs are AABC-ABCC and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (6, -6, -24, 49, 6, -66, 36).

Column 3 of A320525.

Cf. A000392 (oriented), A056327 (unoriented), A304973 (achiral).

I:=[0,0,0,2,10,40,141]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -24*Self(n-3) +49*Self(n-4) +6*Self(n-5) -66*Self(n-6) +36*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018

k=3; Table[(StirlingS2[n,k] - If[EvenQ[n], 2StirlingS2[n/2+1,3] - 2StirlingS2[n/2,3], StirlingS2[(n+3)/2,3] - StirlingS2[(n+1)/2,3]])/2, {n, 1, 30}]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k = 3; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 0, 2, 10, 40,
  141}, 40]

m=40; v=concat([0,0,0,2,10,40,141], vector(m-7)); for(n=8, m, v[n] = 6*v[n-1] -6*v[n-2] -24*v[n-3] +49*v[n-4] +6*v[n-5] -66*v[n-6] +36*v[n-7] ); v \\ G. C. Greubel, Oct 16 2018

a(n) = (S2(n,k) - A(n,k))/2, where k=3 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^3 / Product_{k=1..3} (1 - k*x) - x^3*(1 + 2 x)/((1 - 2 x^2)*(1 - 3 x^2))) / 2.
a(n) = (A000392(n) - A304973(n)) / 2 = A000392(n) - A056327(n) = A056327(n) - A304973(n).

A049434 Stirling numbers of second kind: 8th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053, 20415995028, 189036065010, 1709751003480, 15170932662679, 132511015347084, 1142399079991620, 9741955019900400, 82318282158320505, 690223721118368580, 5749622251945664950

Offset: 8

Wolfdieter Lang

See A000771.

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049435. a(n)= A008277(n, 8).

lst={};Do[f=StirlingS2[n, 8];AppendTo[lst, f], {n, 8, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

G.f.: x^8/product_{k=1..8} (1-k*x).
E.g.f.: ((exp(x)-1)^8)/8!.
a(n) = det(|s(i+8,j+7)|, 1 <= i,j <= n-8), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

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