A205341 -id:A205341

A205337

Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 4.

+10
2

0, 4, 12, 82, 454, 2912, 18652, 124299, 841400, 5800725, 40506816, 286137616, 2040430976, 14670243774, 106225269954, 773958961125, 5670067999156, 41742291894425, 308645064367896, 2291123920091484, 17067970534656790

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Column 4 of A205341.

Number of excursions (walks starting at the origin, ending on the x-axis, and never go below the x-axis in between) with n steps from {-4,-3,-2,-1,1,2,3,4}. - David Nguyen, Dec 20 2016

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

FORMULA

a(n) = Sum_{i=1..n}((Sum_{l=0..i}(binomial(i,l)*(Sum_{j=0..(4*(i-l))/9}((-1)^j*binomial(i-l,j)*binomial(-l+4*(-l-2*j+i)-j+i-1,4*(-l-2*j+i)-j)))*(-1)^l))*a(n-i))/n, a(0)=1. - Vladimir Kruchinin, Apr 07 2017

EXAMPLE

Some solutions for n=5

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..2....3....2....4....4....4....1....2....4....3....3....1....2....3....2....4

..3....5....6....3....0....5....0....4....6....1....5....0....3....1....0....2

..6....1....2....2....1....3....3....6....3....4....3....1....6....2....1....5

..2....2....1....1....3....4....1....4....4....2....4....2....4....3....4....2

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

MATHEMATICA

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Binomial[i, l] Sum[(-1)^j Binomial[i - l, j] Binomial[-l + 4(-l - 2j + i) - j + i - 1, 4(-l - 2j + i) - j], {j, 0, (4(i - l))/9}] (-1)^l, {l, 0, i}] a[n - i], {i, 1, n}]/n];

a /@ Range[1, 21] (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)

PROG

(Maxima)

a(n):=if n=0 then 1 else sum(sum(binomial(i, l)*sum((-1)^j*binomial(i-l, j)*binomial(-l+4*(-l-2*j+i)-j+i-1, 4*(-l-2*j+i)-j), j, 0, (4*(i-l))/9)*(-1)^l, l, 0, i)*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin, Apr 07 2017 */

CROSSREFS

Cf. A205341.

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 26 2012

STATUS

approved

A205336

Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 3.

+10
1

0, 3, 6, 35, 138, 689, 3272, 16522, 83792, 434749, 2278888, 12093271, 64741330, 349470487, 1899418046, 10387322922, 57111322368, 315523027610, 1750681516380, 9751416039535, 54507046599094, 305650440453943, 1718956630038438

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Column 3 of A205341.

Number of excursions (walks starting at the origin, ending on the x-axis, and never go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}. - David Nguyen, Dec 20 2016

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

FORMULA

a(n) = Sum_{i=1..n}((Sum_{l=0..i}(binomial(i,l)*(Sum_{j=0=(3*(i-l))/7}((-1)^j*binomial(i-l,j)*binomial(-l+3*(-l-2*j+i)-j+i-1,3*(-l-2*j+i)-j)))*(-1)^l))*a(n-i))/n, a(0)=1. - Vladimir Kruchinin, Apr 07 2017

EXAMPLE

Some solutions for n=5

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..3....3....3....1....3....1....1....3....3....2....1....3....1....3....3....3

..4....6....2....0....2....3....3....2....5....4....4....1....3....2....2....0

..2....5....5....3....4....4....2....3....4....1....2....2....0....4....0....2

..3....2....2....2....2....1....3....2....2....3....1....1....2....3....2....1

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

MATHEMATICA

a[n_] := a[n] = If[n == 0, 1, Sum[(Sum[Binomial[i, l] (Sum[(-1)^j Binomial[i - l, j] Binomial[-l + 3(-l - 2j + i) - j + i - 1, 3(-l - 2j + i) - j], {j, 0, (3(i - l))/7}]) (-1)^l, {l, 0, i}]) a[n - i], {i, 1, n}]/n];

a /@ Range[1, 23] (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)

PROG

(Maxima)

a(n):=if n=0 then 1 else sum((sum(binomial(i, l)*(sum((-1)^j*binomial(i-l, j)*binomial(-l+3*(-l-2*j+i)-j+i-1, 3*(-l-2*j+i)-j), j, 0, (3*(i-l))/7))*(-1)^l, l, 0, i))*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin, Apr 07 2017 */

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 26 2012

STATUS

approved

A205338

Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 5.

+10
1

0, 5, 20, 160, 1130, 8927, 71630, 594405, 5025740, 43243674, 377127756, 3327001441, 29634744950, 266164547110, 2407763862342, 21918167505714, 200631620380132, 1845576127894008, 17052050519557200, 158176470846492722

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Column 5 of A205341.

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

FORMULA

a(n) = Sum_{i=1..n}((Sum_{l=0..i}(binomial(i,l)*(Sum_{j=0..(5*(i-l))/11}((-1)^j*binomial(i-l,j)*binomial(-l+5*(-l-2*j+i)-j+i-1,5*(-l-2*j+i)-j)))*(-1)^l))*a(n-i))/n, a(0)=1. - Vladimir Kruchinin, Apr 07 2017

EXAMPLE

Some solutions for n=5:

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..5....5....5....2....5....5....4....2....3....2....2....2....4....1....3....1

..8....6...10....3....8....4....6....1....5....5....6....5....7....4....1....4

..3....9....9....0....5....0....2....5....0....4....1....3....4....5....2....7

..1....4....5....3....2....2....3....2....1....3....4....4....2....3....1....3

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

PROG

(Maxima)

a(n):=if n=0 then 1 else sum((sum(binomial(i, l)*(sum((-1)^j*binomial(i-l, j)*binomial(-l+5*(-l-2*j+i)-j+i-1, 5*(-l-2*j+i)-j), j, 0, (5*(i-l))/11))*(-1)^l, l, 0, i))*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin, Apr 07 2017 */

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 26 2012

STATUS

approved

A205339

Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 6

+10
1

0, 6, 30, 277, 2370, 22297, 214724, 2133784, 21632020, 223143400, 2333651994, 24689732388, 263770658256, 2841616524516, 30835061022020, 336721385300276, 3697585562072924, 40805356360923728, 452314009660461816

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Column 6 of A205341

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

EXAMPLE

Some solutions for n=5

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..3....1....5....3....6....2....5....2....3....5....5....3....3....3....3....5

..1....3....4....1....4....5....8....4....5....2....9....2....0....7....1....4

..5....9....8....0....2....0....7....3....0....3....8....7....6...10....4....9

..2....6....5....2....3....4....6....5....2....5....4....5....1....5....5....5

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

KEYWORD

nonn

AUTHOR

R. H. Hardin Jan 26 2012

STATUS

approved

A205340

Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 7

+10
1

0, 7, 42, 441, 4424, 48335, 542850, 6285127, 74286702, 893407361, 10894937088, 134418087923, 1674757658798, 21042485711561, 266318361927208, 3392084001234202, 43447635519011920, 559277626577030221

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Column 7 of A205341

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

EXAMPLE

Some solutions for n=5

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..4....4....5....7....2....5....6....1....2....6....4....4....2....7....4....4

..5....5...10....9....7...12....8....0....8....4....9...11....8....1....9....5

.11...12....4...11....0...14....9....1...10....8....3....4...13....5....7....0

..6....6....5....5....6....7....4....4....4....7....7....5....7....2....1....4

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

KEYWORD

nonn

AUTHOR

R. H. Hardin Jan 26 2012

STATUS

approved

A205342

Number of length 5 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

+10
1

2, 11, 35, 82, 160, 277, 441, 660, 942, 1295, 1727, 2246, 2860, 3577, 4405, 5352, 6426, 7635, 8987, 10490, 12152, 13981, 15985, 18172, 20550, 23127, 25911, 28910, 32132, 35585, 39277, 43216, 47410, 51867, 56595, 61602, 66896, 72485, 78377, 84580, 91102

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Row 4 of A205341.

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

FORMULA

Empirical: a(n) = (4/3)*n^3 - (1/2)*n^2 + (7/6)*n.

Conjectures from Colin Barker, Jun 11 2018: (Start)

G.f.: x*(2 + 3*x + 3*x^2) / (1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.

(End)

EXAMPLE

Some solutions for n=5:

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..2....5....3....2....2....1....5....2....5....4....1....5....4....4....5....2

..0....8....6....4....6....6....1....7....1....9....6....7....5....2....8....5

..3....3....5....1....5....2....2....3....5....4....5....4....2....4....5....4

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

CROSSREFS

Cf. A205341.

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 26 2012

STATUS

approved

A205343

Number of length 6 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

+10
1

0, 24, 138, 454, 1130, 2370, 4424, 7588, 12204, 18660, 27390, 38874, 53638, 72254, 95340, 123560, 157624, 198288, 246354, 302670, 368130, 443674, 530288, 629004, 740900, 867100, 1008774, 1167138, 1343454, 1539030, 1755220, 1993424, 2255088

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Row 5 of A205341.

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

FORMULA

Empirical: a(n) = (23/12)*n^4 - (1/2)*n^3 + (1/12)*n^2 - (3/2)*n.

Conjectures from Colin Barker, Jun 11 2018: (Start)

G.f.: 2*x^2*(12 + 9*x + 2*x^2) / (1 - x)^5.

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)

EXAMPLE

Some solutions for n=5:

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..5....4....3....4....2....4....3....5....5....3....5....4....5....1....1....1

..1....2....2....9....5....7....6....6....2....6....0....2....2....0....6....5

..5....0....5....8....1....3....7....8....6....4....5....0....1....2....9....9

..3....4....1....5....5....1....2....4....2....5....2....1....5....1....5....4

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

CROSSREFS

Cf. A205341.

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 26 2012

STATUS

approved

A205344

Number of length 7 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

+10
1

5, 93, 689, 2912, 8927, 22297, 48335, 94456, 170529, 289229, 466389, 721352, 1077323, 1561721, 2206531, 3048656, 4130269, 5499165, 7209113, 9320208, 11899223, 15019961, 18763607, 23219080, 28483385, 34661965, 41869053, 50228024

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Row 6 of A205341.

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

FORMULA

Empirical: a(n) = (44/15)*n^5 - (5/12)*n^4 + (5/12)*n^2 + (31/15)*n.

Conjectures from Colin Barker, Jun 11 2018: (Start)

G.f.: x*(5 + 63*x + 206*x^2 + 73*x^3 + 5*x^4) / (1 - x)^6.

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)

EXAMPLE

Some solutions for n=5:

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..1....1....5....3....5....1....1....5....4....3....5....4....5....1....3....4

..6....4...10....6....0....2....4....3....8....6...10....3....8....5....1....7

..8....1....5....1....2....0....8....6....7....5....6....6....7....1....5....6

..7....4....3....5....4....4....6....5....3....3....2....7....5....5....3....7

..4....3....5....2....1....3....1....2....2....1....4....5....1....2....2....3

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

CROSSREFS

Cf. A205341.

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 26 2012

STATUS

approved

A205345

Number of length 8 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

+10
1

0, 272, 3272, 18652, 71630, 214724, 542850, 1211784, 2459988, 4633800, 8215988, 13857668, 22413586, 34980764, 52940510, 78003792, 112259976, 158228928, 218916480, 297873260, 399256886, 527897524, 689366810, 890050136, 1137222300

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Row 7 of A205341.

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

FORMULA

Empirical: a(n) = (841/180)*n^6 - (1/3)*n^5 - (19/36)*n^4 + (1/3)*n^3 - (103/90)*n^2 - 3*n.

Conjectures from Colin Barker, Jun 11 2018: (Start)

G.f.: 2*x^2*(136 + 684*x + 730*x^2 + 129*x^3 + 3*x^4) / (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)

EXAMPLE

Some solutions for n=5:

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..5....3....5....2....2....2....2....3....5....3....1....2....4....2....3....4

..6....7....8....6....3....4....4....4....9....5....2....5....8....6....5....7

..7....8...11....7....4....3....2....7...12....1....0....4....7....4....6....8

.10....4....9....6....3....6....7....3....7....2....4....3....3....8....3....7

..6....5....4....4....0....5....3....6....3....7....6....0....1....3....1....8

..5....3....2....3....3....4....4....2....5....2....5....2....5....5....3....3

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

CROSSREFS

Cf. A205341.

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 26 2012

STATUS

approved