A205341 -id:A205341 - cp's OEIS frontend

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.

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

Offset: 1

R. H. Hardin, Jan 26 2012

nonn

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

			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

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.

Cf. A205341.

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 *)

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 */

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

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.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jan 26 2012

nonn

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

			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

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.

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 *)

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 */

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

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.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jan 26 2012

nonn

Column 5 of A205341.

			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

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

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 */

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

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.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin Jan 26 2012

nonn

Column 6 of A205341

			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

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

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.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin Jan 26 2012

nonn

Column 7 of A205341

			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

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

A205342 Number of length 5 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jan 26 2012

nonn

Row 4 of A205341.

			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

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

Cf. A205341.

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)

A205343 Number of length 6 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jan 26 2012

nonn

Row 5 of A205341.

			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

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

Cf. A205341.

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)

A205344 Number of length 7 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jan 26 2012

nonn

Row 6 of A205341.

			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

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

Cf. A205341.

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)

A205345 Number of length 8 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jan 26 2012

nonn

Row 7 of A205341.

			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

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

Cf. A205341.

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)

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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.

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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.

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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.

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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.

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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.

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A205342 Number of length 5 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

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A205343 Number of length 6 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

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A205344 Number of length 7 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

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A205345 Number of length 8 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than n.

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