A200785 -id:A200785 - cp's OEIS frontend

A200781 G.f.: 1/(1-5x+10x^3-5*x^4).

1, 5, 25, 115, 530, 2425, 11100, 50775, 232275, 1062500, 4860250, 22232375, 101698250, 465201250, 2127983750, 9734098125, 44526969375, 203681015625, 931704015625, 4261920875000, 19495429065625, 89178510250000, 407931862578125, 1866014626609375, 8535765175875000, 39045399804843750, 178606512071015625, 817004981729375000

Offset: 0

R. H. Hardin, Nov 22 2011

nonn

Number of words of length n over an alphabet of size 5 which do not contain any strictly decreasing factor (consecutive subword) of length 3. For alphabets of size 2, 3, 4, 6 see A000079, A076264, A072335, A200782.

Equivalently, number of 0..4 arrays x(0..n-1) of n elements without any two consecutive increases.

			Some solutions for n=5:
..1....3....4....0....1....0....4....0....2....1....4....1....2....2....4....4
..3....4....4....2....1....0....3....3....1....4....1....1....4....4....3....3
..3....1....0....2....0....2....0....3....3....0....4....3....0....1....4....4
..2....0....2....4....4....0....3....2....0....0....3....2....0....2....1....3
..4....4....2....2....0....3....3....2....1....0....4....1....3....1....0....2

R. H. Hardin and N. J. A. Sloane, Table of n, a(n) for n = 0..249 [The first 210 terms were computed by R. H. Hardin]
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.
Index entries for linear recurrences with constant coefficients, signature (5,0,-10,5).

The g.f. corresponds to row 5 of triangle A225682.
Column 4 of A200785.
Cf. A076264, A072335, A200782.

Vec(1/(1-5*x+10*x^3-5*x^4) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

a(n) = 5*a(n-1) - 10*a(n-3) + 5*a(n-4).

Edited by N. J. A. Sloane, May 21 2013

A200782 Expansion of 1 / (1 - 6x + 20x^3 - 15*x^4 + x^6).

Original entry on oeis.org

1, 6, 36, 196, 1071, 5796, 31395, 169884, 919413, 4975322, 26924106, 145698840, 788446400, 4266656226, 23088902733, 124944995676, 676136621430, 3658895818470, 19800020091895, 107147296401684, 579824822459421, 3137707025200000

Offset: 0

R. H. Hardin, Nov 22 2011

a(n) is the number of words of length n over an alphabet of size 6 which do not contain any strictly decreasing factor (consecutive subword) of length 3.
Equivalently, dimensions of homogeneous components of the universal associative envelope for a certain nonassociative triple system [Bremner].
This is the g.f. corresponding to row 6 of A225682.

			a(n) is also the number of words of length n over an alphabet of size 6 which do not contain any strictly increasing factor of length 3. Some solutions for n=5:
..5....5....0....3....2....4....3....3....3....3....0....3....3....1....0....1
..1....5....0....0....4....5....1....1....3....5....1....0....2....0....3....4
..3....5....1....0....4....3....1....4....5....0....1....5....1....0....0....3
..0....0....0....4....1....1....1....4....2....4....1....1....2....5....4....1
..1....4....2....0....0....0....1....3....1....4....3....2....2....2....4....5

R. H. Hardin and N. J. Sloane, Table of n, a(n) for n = 0..239 [The first 210 terms were computed by R. H. Hardin]
M. R. Bremner, Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures, arXiv:1303.0920 [math.RA], 2013
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.
Index entries for linear recurrences with constant coefficients, signature (6,0,-20,15,0,-1).

Column 5 of A200785.

G.f. corresponds to row 6 of A225682.

CoefficientList[Series[1 / (1 - 6*x + 20*x^3 - 15*x^4 + x^6), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 26 2015 *)
LinearRecurrence[{6,0,-20,15,0,-1},{1,6,36,196,1071,5796},30] (* Harvey P. Dale, Jul 28 2019 *)

Vec(1/(1-6*x+20*x^3-15*x^4+x^6) + O(x^30)) \\ Michel Marcus, Jan 26 2015

G.f.: 1 / (1 - 6*x + 20*x^3 - 15*x^4 + x^6).

a(n) = 6*a(n-1) - 20*a(n-3) + 15*a(n-4) - a(n-6).

Entry revised by N. J. A. Sloane, May 17 2013, merging this with A225381

Typo in name corrected by Michel Marcus, Jan 26 2015

A200783 G.f.: 1/(1-7x+35x^3-35x^4+7x^6-x^7).

Original entry on oeis.org

1, 7, 49, 308, 1946, 12152, 75992, 474566, 2964416, 18514405, 115637431, 722234149, 4510869636, 28173535572, 175963587528, 1099016234232, 6864129384252, 42871313869692, 267761500599901, 1672358840069239, 10445056851917149, 65236724277810632, 407449213173792062, 2544806826734163992, 15894107968042546424, 99269879914558590146

Offset: 0

R. H. Hardin Nov 22 2011

nonn

Number of words of length n over an alphabet of size 7 which do not contain any strictly decreasing factor (consecutive subword) of length 3.

Number of 0..6 arrays x(0..n-1) of n elements without any two consecutive increases.

			Some solutions for n=5
..6....2....6....3....4....4....6....6....5....3....2....4....5....0....5....5
..4....5....0....4....1....6....4....5....1....1....2....6....6....6....3....6
..4....4....0....4....5....3....5....5....5....1....5....3....3....6....4....2
..3....6....2....5....5....2....2....4....5....5....3....3....2....1....4....5
..4....5....0....3....1....0....4....3....5....5....2....1....0....0....5....3

R. H. Hardin and N. J. Sloane, Table of n, a(n) for n = 0..249 [The first 210 terms were computed by R. H. Hardin]
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.
Index entries for linear recurrences with constant coefficients, signature (7, 0, -35, 35, 0, -7, 1).

Column 6 of A200785.

G.f. corresponds to row 7 of A225682.

CoefficientList[Series[1/(1-7x+35x^3-35x^4+7x^6-x^7),{x,0,30}],x] (* or *) LinearRecurrence[{7,0,-35,35,0,-7,1},{1,7,49,308,1946,12152,75992},30] (* Harvey P. Dale, Jul 23 2014 *)

a(n) = 7*a(n-1) - 35*a(n-3) + 35*a(n-4) - 7*a(n-6) + a(n-7).

Edited by N. J. A. Sloane, May 21 2013

A200784 Number of 0..7 arrays x(0..n+1) of n+2 elements without any two consecutive increases.

Original entry on oeis.org

456, 3270, 23136, 164004, 1160616, 8216484, 58154912, 411637168, 2913595712, 20622837480, 145970677056, 1033197881712, 7313093248992, 51762926098992, 366383987227392, 2593308396911680, 18355737644921600, 129924040926296800

Offset: 1

R. H. Hardin, Nov 22 2011

nonn

Column 7 of A200785.

			Some solutions for n=3
..7....5....7....4....5....2....3....7....5....5....5....6....1....5....4....4
..0....5....4....2....3....2....1....4....0....5....7....4....0....5....4....4
..0....5....4....1....6....6....3....5....0....4....6....6....6....7....4....4
..0....1....7....1....5....5....0....0....6....1....6....3....0....0....4....7
..1....4....0....6....5....2....6....5....3....4....2....4....1....2....4....3

R. H. Hardin, Table of n, a(n) for n = 1..210
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.

Empirical: a(n) = 8*a(n-1) -56*a(n-3) +70*a(n-4) -28*a(n-6) +8*a(n-7).

Empirical g.f.: 2*x*(228 - 189*x - 1512*x^2 + 2226*x^3 - 108*x^4 - 864*x^5 + 256*x^6) / ((1 - 2*x)*(1 + 2*x - 2*x^2)*(1 - 8*x + 6*x^2 + 4*x^3 - 2*x^4)). - Colin Barker, Oct 15 2017

A200786 Number of 0..n arrays x(0..3) of 4 elements without any two consecutive increases.

Original entry on oeis.org

16, 75, 225, 530, 1071, 1946, 3270, 5175, 7810, 11341, 15951, 21840, 29225, 38340, 49436, 62781, 78660, 97375, 119245, 144606, 173811, 207230, 245250, 288275, 336726, 391041, 451675, 519100, 593805, 676296, 767096, 866745, 975800, 1094835

Offset: 1

R. H. Hardin, Nov 22 2011

nonn

Row 2 of A200785.

			Some solutions for n=3
..0....1....0....3....2....3....3....2....3....1....0....1....0....3....1....0
..0....3....3....1....2....3....2....0....3....3....3....2....0....2....1....2
..3....2....3....1....1....0....2....3....3....1....1....1....2....1....3....1
..1....2....1....2....3....0....1....0....3....2....2....2....2....2....1....1

R. H. Hardin, Table of n, a(n) for n = 1..140
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.

Empirical: a(n) = (17/24)*n^4 + (43/12)*n^3 + (151/24)*n^2 + (53/12)*n + 1.
Conjectures from Colin Barker, Oct 15 2017: (Start)
G.f.: x*(16 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = (24 + 106*n + 151*n^2 + 86*n^3 + 17*n^4) / 24.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A200787 Number of 0..n arrays x(0..4) of 5 elements without any two consecutive increases.

Original entry on oeis.org

32, 216, 840, 2425, 5796, 12152, 23136, 40905, 68200, 108416, 165672, 244881, 351820, 493200, 676736, 911217, 1206576, 1573960, 2025800, 2575881, 3239412, 4033096, 4975200, 6085625, 7385976, 8899632, 10651816, 12669665, 14982300

Offset: 1

R. H. Hardin, Nov 22 2011

nonn

Row 3 of A200785.

			Some solutions for n=3
..2....1....3....2....2....3....1....3....1....1....2....1....0....1....2....0
..0....0....3....3....1....2....3....1....3....2....0....0....3....3....2....0
..0....1....1....3....3....1....3....1....0....0....0....2....3....2....3....0
..3....1....1....1....1....0....3....2....3....3....3....2....1....1....0....0
..0....1....1....0....2....3....3....1....1....0....1....0....3....1....2....3

R. H. Hardin, Table of n, a(n) for n = 1..139

Empirical: a(n) = (7/12)*n^5 + (47/12)*n^4 + (39/4)*n^3 + (133/12)*n^2 + (17/3)*n + 1.
Conjectures from Colin Barker, Oct 15 2017: (Start)
G.f.: x*(32 + 24*x + 24*x^2 - 15*x^3 + 6*x^4 - x^5) / (1 - x)^6.
a(n) = (1 + n)^2*(12 + 44*n + 33*n^2 + 7*n^3) / 12 .
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A200788 Number of 0..n arrays x(0..5) of 6 elements without any two consecutive increases.

Original entry on oeis.org

64, 622, 3136, 11100, 31395, 75992, 164004, 324087, 597190, 1039654, 1726660, 2756026, 4252353, 6371520, 9305528, 13287693, 18598188, 25569934, 34594840, 46130392, 60706591, 78933240, 101507580, 129222275, 162973746, 203770854

Offset: 1

R. H. Hardin, Nov 22 2011

nonn

Row 4 of A200785.

			Some solutions for n=3
..2....3....3....2....3....2....1....1....3....3....1....1....2....3....2....1
..3....2....1....3....0....0....3....0....0....3....3....1....2....3....2....1
..0....3....3....3....0....1....3....0....1....0....3....3....1....0....2....1
..3....3....2....3....3....1....0....2....1....2....0....0....0....2....2....2
..3....1....2....3....0....1....1....1....3....0....2....0....3....2....1....2
..3....3....1....2....0....2....1....2....3....0....2....2....1....2....2....3

R. H. Hardin, Table of n, a(n) for n = 1..138

Empirical: a(n) = (349/720)*n^6 + (321/80)*n^5 + (1883/144)*n^4 + (1013/48)*n^3 + (3139/180)*n^2 + (413/60)*n + 1.
Conjectures from Colin Barker, Oct 15 2017: (Start)
G.f.: x*(64 + 174*x + 126*x^2 - 30*x^3 + 21*x^4 - 7*x^5 + x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A200789 Number of 0..n arrays x(0..6) of 7 elements without any two consecutive increases.

Original entry on oeis.org

128, 1791, 11704, 50775, 169884, 474566, 1160616, 2562633, 5217520, 9944957, 17946864, 30927871, 51238812, 82045260, 127523120, 193083297, 285627456, 413836891, 588496520, 822856023, 1133030140, 1538440146, 2062298520

Offset: 1

R. H. Hardin, Nov 22 2011

nonn

Row 5 of A200785.

			Some solutions for n=3
..1....0....2....0....0....3....0....1....0....2....2....2....0....3....1....0
..2....2....3....0....2....3....3....1....3....2....0....2....2....1....3....3
..0....0....3....1....2....2....0....0....2....0....0....2....0....1....3....2
..1....1....0....1....3....1....1....0....3....2....3....0....0....0....2....1
..0....1....3....0....1....3....0....0....1....1....2....3....2....3....0....0
..1....3....0....0....1....3....0....1....1....3....3....2....2....0....2....0
..1....1....0....2....1....1....0....0....3....0....0....2....0....2....1....3

R. H. Hardin, Table of n, a(n) for n = 1..137

Empirical: a(n) = (2017/5040)*n^7 + (1427/360)*n^6 + (5759/360)*n^5 + (607/18)*n^4 + (28459/720)*n^3 + (9113/360)*n^2 + (848/105)*n + 1.
Conjectures from Colin Barker, Oct 15 2017: (Start)
The formulas below are consistent with the conjectured formula above.
G.f.: x*(128 + 767*x + 960*x^2 + 123*x^3 + 60*x^4 - 28*x^5 + 8*x^6 - x^7) / (1 - x)^8.
a(n) = (5040 + 40704*n + 127582*n^2 + 199213*n^3 + 169960*n^4 + 80626*n^5 + 19978*n^6 + 2017*n^7) / 5040.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

A200790 Number of 0..n arrays x(0..7) of 8 elements without any two consecutive increases.

Original entry on oeis.org

256, 5157, 43681, 232275, 919413, 2964416, 8216484, 20273247, 45611500, 95196145, 186686721, 347374261, 617994573, 1057577400, 1749504272, 2808961221, 4391985888, 6706321909, 10024306825, 14698033119, 21177035341, 30028769640

Offset: 1

R. H. Hardin, Nov 22 2011

nonn

Row 6 of A200785.

			Some solutions for n=3
..3....1....2....2....0....2....1....2....2....3....0....0....0....3....3....1
..0....1....0....2....2....3....3....2....1....3....0....1....1....0....0....3
..0....3....3....1....0....1....3....1....2....0....3....0....1....0....3....1
..1....3....2....2....3....3....2....2....0....2....0....0....1....3....3....2
..0....1....3....2....1....3....3....1....0....1....2....3....3....1....0....0
..3....0....2....2....3....0....3....3....3....3....1....2....0....2....0....3
..0....1....1....2....0....0....2....0....1....0....3....1....2....1....3....1
..3....0....3....3....3....1....3....0....0....3....1....0....2....3....1....0

R. H. Hardin, Table of n, a(n) for n = 1..136

Empirical: a(n) = (6679/20160)*n^8 + (4799/1260)*n^7 + (26449/1440)*n^6 + (2162/45)*n^5 + (212153/2880)*n^4 + (6019/90)*n^3 + (174571/5040)*n^2 + (3893/420)*n + 1.
Conjectures from Colin Barker, Oct 15 2017: (Start)
G.f.: x*(256 + 2853*x + 6484*x^2 + 3294*x^3 + 522*x^4 - 79*x^5 + 36*x^6 - 9*x^7 + x^8) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>9.
(End)

A200791 Number of 0..n arrays x(0..8) of 9 elements without any two consecutive increases.

Original entry on oeis.org

512, 14849, 163020, 1062500, 4975322, 18514405, 58154912, 160338680, 398601390, 910893148, 1941103528, 3899741885, 7449762880, 13624665670, 23987233104, 40838614531, 67488892468, 108601809395, 170627966340, 262342539690

Offset: 1

R. H. Hardin, Nov 22 2011

nonn

Row 7 of A200785.

			Some solutions for n=3
..0....0....0....0....0....1....1....1....3....0....1....0....1....1....3....1
..0....0....0....3....3....3....3....3....0....3....3....3....0....1....1....3
..3....0....2....0....1....0....0....1....0....2....3....3....2....2....0....1
..2....1....0....3....0....3....0....3....1....2....1....3....1....2....1....0
..2....1....3....3....0....0....2....3....0....3....1....1....0....3....1....0
..3....3....2....0....3....0....0....1....2....3....3....0....3....2....3....2
..2....2....2....3....2....2....0....1....1....1....2....1....1....0....1....1
..1....3....3....2....3....0....0....3....1....0....2....1....0....1....0....0
..3....0....1....2....3....1....3....2....3....2....2....0....0....0....3....2

R. H. Hardin, Table of n, a(n) for n = 1..135

Empirical: a(n) = (99377/362880)*n^9 + (48247/13440)*n^8 + (243673/12096)*n^7 + (60529/960)*n^6 + (2076437/17280)*n^5 + (274529/1920)*n^4 + (952027/9072)*n^3 + (152461/3360)*n^2 + (26399/2520)*n + 1.
Conjectures from Colin Barker, Oct 15 2017: (Start)
G.f.: x*(512 + 9729*x + 37570*x^2 + 39065*x^3 + 11862*x^4 + 551*x^5 + 124*x^6 - 45*x^7 + 10*x^8 - x^9) / (1 - x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>10.
(End)

cp's OEIS Frontend

A200781 G.f.: 1/(1-5*x+10*x^3-5*x^4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A200782 Expansion of 1 / (1 - 6*x + 20*x^3 - 15*x^4 + x^6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A200783 G.f.: 1/(1-7*x+35*x^3-35*x^4+7*x^6-x^7).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A200784 Number of 0..7 arrays x(0..n+1) of n+2 elements without any two consecutive increases.

Views

Author

Keywords

Comments

Examples

Links

Formula

A200786 Number of 0..n arrays x(0..3) of 4 elements without any two consecutive increases.

Views

Author

Keywords

Comments

Examples

Links

Formula

A200787 Number of 0..n arrays x(0..4) of 5 elements without any two consecutive increases.

Views

Author

Keywords

Comments

Examples

Links

Formula

A200788 Number of 0..n arrays x(0..5) of 6 elements without any two consecutive increases.

Views

Author

Keywords

Comments

Examples

Links

Formula

A200789 Number of 0..n arrays x(0..6) of 7 elements without any two consecutive increases.

Views

Author

Keywords

Comments

Examples

Links

Formula

A200790 Number of 0..n arrays x(0..7) of 8 elements without any two consecutive increases.

Views

A200781 G.f.: 1/(1-5x+10x^3-5*x^4).

A200782 Expansion of 1 / (1 - 6x + 20x^3 - 15*x^4 + x^6).

A200783 G.f.: 1/(1-7x+35x^3-35x^4+7x^6-x^7).