A007695 -id:A007695 - cp's OEIS frontend

A011828 Number of f-vectors for simplicial complexes of dimension at most 3 on at most n-1 vertices.

2, 3, 5, 10, 26, 95, 457, 2246, 9705, 35926, 115688, 331201, 859587, 2054860, 4582126, 9627831, 19217260, 36679253, 67308375, 119286676, 204940824, 342425909, 557944719, 888630900, 1386246251, 2121866592, 3191757298

Offset: 1

Svante Linusson (linusson(AT)math.kth.se)

nonn

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3 (p. 743).
S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.

Cf. A011827, A007695.

a(n+1) = (12*n^10 -112*n^9 +351*n^8 -132*n^7 +378*n^6 -2856*n^5 +4839*n^4 +56812*n^3 -5580*n^2 +309168*n +725760)/362880 fits terms up to 3191757298. [Frank Ellermann]

Empirical G.f.: -x*(x^10 -11*x^9 +69*x^8 -130*x^7 +380*x^6 -400*x^5 +356*x^4 -210*x^3 +82*x^2 -19*x +2)/(x -1)^11. [Colin Barker, Sep 18 2012]

A011833 Number of f-vectors for simplicial complexes of dimension at most 8 on at most n-1 vertices.

Original entry on oeis.org

2, 3, 5, 10, 26, 96, 553, 5461, 100709, 3718354, 289725508, 49224068017, 13203989190379, 2498210880028303, 272472900722736282, 18445455348353453707, 846271790301299050986, 28189314873080070171124

Offset: 1

Svante Linusson (linusson(AT)math.kth.se)

nonn

See A007695 for further information.

A011829 Number of f-vectors for simplicial complexes of dimension at most 4 on at most n-1 vertices.

Original entry on oeis.org

2, 3, 5, 10, 26, 96, 552, 4908, 48230, 398663, 2631241, 14192097, 64663638, 256174350, 902972232, 2883651027, 8463753978, 23094833355, 59133598085, 143164108028, 329810868994, 726833391860, 1539215246944, 3144340388550

Offset: 1

Svante Linusson (linusson(AT)math.kth.se)

nonn

Apparently a polynomial (n^15)/32659200 + .... [Frank Ellermann]

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3 (p. 743).
S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.

Cf. A011828, A011830, A011831, A011832, A011833, A007695.

Empirical G.f.: -x*(x^15 -16*x^14 +114*x^13 -1154*x^12 -2541*x^11 -16919*x^10 -6585*x^9 -17282*x^8 +9460*x^7 -8548*x^6 +5196*x^5 -2426*x^4 +830*x^3 -197*x^2 +29*x -2)/(x -1)^16. [Colin Barker, Sep 18 2012]

A220880 Number of profiles of monotone Boolean functions of n variables.

Original entry on oeis.org

1, 2, 4, 9, 25, 95, 552, 5460, 100708, 3718353, 289725508, 49513793525, 19089032278260, 16951604697397301, 35231087224279091309, 173550485517380958360610, 2047581288200721764035942913

Offset: 0

N. J. A. Sloane, Dec 28 2012

nonn

Equals A007695(n) - 1.

Matthias Thimm, On the expressivity of inconsistency measures, Artificial Intelligence, Volume 234, May 2016, Pages 120-151.

Tamon Stephen and Timothy Yusun, Counting inequivalent monotone Boolean functions, arXiv preprint arXiv:1209.4623 [cs.DS], 2012.

Cf. A007695.

A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 10, 5, 5, 1, 1, 15, 15, 6, 6, 1, 1, 35, 21, 21, 7, 7, 1, 1, 56, 56, 28, 28, 8, 8, 1, 1, 126, 84, 84, 36, 36, 9, 9, 1, 1, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1, 792, 792, 495, 495, 220, 220, 66, 66, 12, 12, 1, 1

Offset: 1

Stefano Spezia, May 31 2020

T(n, k) is a tight upper bound of the cardinality of an intersecting Sperner family or antichain of the set {1, 2,..., n}, where every collection of pairwise independent subsets is characterized by an intersection of cardinality at least k (see Theorem 1.3 in Wong and Tay).

Equals A061554 with the first row of the array (resp. the first column of the triangle) removed. - Georg Fischer, Jul 26 2023

			The triangle T(n, k) begins
n\k|  1   2   3   4   5   6   7   8
---+-------------------------------
1  |  1
2  |  1   1
3  |  3   1   1
4  |  4   4   1   1
5  | 10   5   5   1   1
6  | 15  15   6   6   1   1
7  | 35  21  21   7   7   1   1
8  | 56  56  28  28   8   8   1   1
...

Eric Charles Milner, A Combinatorial Theorem On Systems of Sets, Journal of the London Mathematical Society, 43, (1968), 204-206.
W. H. W. Wong and E. G. Tay, On Cross-intersecting Sperner Families, arXiv:2001.01910 [math.CO], 2020.
Index entries for sequences related to antichains.

Cf. A000372, A001405, A004526, A007318, A007695, A061554, A266696, A325982, A325983.

Cf. A037951 (k=3), A037952 (k=1), A037953 (k=5), A037954 (k=7), A037955 (k=2), A037956 (k=4), A037957 (k=6), A037958 (k=8), A045621 (row sums).

T[n_,k_]:=Binomial[n,Floor[(n+k+1)/2]]; Table[T[n,k],{n,12},{k,n}]//Flatten

T(n, k) = binomial(n, (n+k+1)\2);
vector(10, n, vector(n, k, T(n, k))) \\ Michel Marcus, Jun 01 2020

T(n, k) = A007318(n, A004526(n+k+1)) with k <= n.

A088505 a(n) = (2^(3*n-1))/(integral_{x=0..1} (1-x^4)^n dx).

Original entry on oeis.org

5, 45, 390, 3315, 27846, 232050, 1922700, 15862275, 130423150, 1069469830, 8750207700, 71460029550, 582674087100, 4744631852100, 38589672397080, 313541088226275, 2545215892660350, 20644528907133950, 167329339563085700

Offset: 1

Al Hakanson (hawkuu(AT)excite.com), Nov 13 2003

nonn

			a(3)=390 (a(0) would be 1/2, so the sequence begins at n=1).

f[n_] := 2^(3n - 1)/Integrate[(1 - x^4)^n, {x, 0, 1}]; Table[ f[n], {n, 1, 19}] (* Robert G. Wilson v, Feb 26 2004 *)

a(n)=round(2^(3*n-1)/(n!*Pi*sqrt(2)/(4*gamma(3/4)*gamma(n+5/4))))

The integral is equal to n!*Pi*sqrt(2)/(4*GAMMA(3/4)*GAMMA(n+5/4)). - N. J. A. Sloane
GAMMA(3/4)*GAMMA(n+5/4) is Pi*sqrt(2)*A007696(n+1)/4^(n+1), so the integral is n!*4^n/A007695(n+1) and a(n) = 2^(n-1)*A007696(n+1)/n!. - R. J. Mathar, Feb 04 2021
D-finite with recurrence n*a(n) +2*(-4*n-1)*a(n-1)=0. - R. J. Mathar, Feb 04 2021

More terms from Benoit Cloitre, Nov 14 2003

cp's OEIS Frontend

A011828 Number of f-vectors for simplicial complexes of dimension at most 3 on at most n-1 vertices.

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A011833 Number of f-vectors for simplicial complexes of dimension at most 8 on at most n-1 vertices.

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A011829 Number of f-vectors for simplicial complexes of dimension at most 4 on at most n-1 vertices.

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A220880 Number of profiles of monotone Boolean functions of n variables.

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A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

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A088505 a(n) = (2^(3*n-1))/(integral_{x=0..1} (1-x^4)^n dx).

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