A278396 -id:A278396 - cp's OEIS frontend

A278391 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-2,-1,0,1,2}.

1, 2, 7, 29, 126, 565, 2583, 11971, 56038, 264345, 1254579, 5983628, 28655047, 137697549, 663621741, 3206344672, 15525816066, 75324830665, 366071485943, 1781794374016, 8684511754535, 42381025041490, 207055067487165, 1012617403658500, 4956924278927910

Offset: 0

David Nguyen, Nov 20 2016

By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.

Andrew Howroyd, Table of n, a(n) for n = 0..200
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:1609.06473 [math.CO], 2016.

Cf. A276903, A278392, A278393, A278394, A278395, A278396, A278398.

frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -2, 2}]; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
seq[25] (* Jean-François Alcover, Jun 30 2018, after Andrew Howroyd *)

seq(n)={my(v=vector(n), m=sum(i=-2, 2, x^i), p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018

A278394 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-2,-1,1,2}.

Original entry on oeis.org

1, 2, 5, 17, 58, 209, 761, 2823, 10557, 39833, 151147, 576564, 2208163, 8486987, 32714813, 126430229, 489685674, 1900350201, 7387530575, 28763059410, 112142791763, 437774109384, 1710883748796, 6693281604018, 26210038447737, 102724200946467, 402925631267151

Offset: 0

David Nguyen, Nov 20 2016

By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.

Andrew Howroyd, Table of n, a(n) for n = 0..200
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:1609.06473 [math.CO], 2016.

Cf. A276852, A278391, A278392, A278393, A278395, A278396, A278398.

frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -2, 2}] - 1; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
seq[27] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

seq(n)={my(v=vector(n), m=sum(i=-2, 2, x^i)-1, p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018

A278392 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,0,1,2,3}.

Original entry on oeis.org

1, 3, 15, 87, 530, 3329, 21316, 138345, 906853, 5989967, 39804817, 265812731, 1782288408, 11991201709, 80911836411, 547334588037, 3710610424765, 25204313298581, 171492983631249, 1168638213247713, 7974592724571446, 54484621312318007, 372671912259214487

Offset: 0

David Nguyen, Nov 20 2016

By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.

Andrew Howroyd, Table of n, a(n) for n = 0..200
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:1609.06473 [math.CO], 2016.

Cf. A276852, A278391, A278393, A278394, A278395, A278396, A278398.

frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}]; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
seq[23] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

seq(n)={my(v=vector(n), m=sum(i=-3, 3, x^i), p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018

A278393 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-4,-3,-2,-1,0,1,2,3,4}.

Original entry on oeis.org

1, 4, 26, 194, 1521, 12289, 101205, 844711, 7120398, 60477913, 516774114, 4437360897, 38256405777, 330948944639, 2871299293535, 24973776734091, 217690276938940, 1901204163460913, 16632544424086901, 145730139895667887, 1278596503973570665, 11231908572986043199

Offset: 0

David Nguyen, Nov 20 2016

By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.

Andrew Howroyd, Table of n, a(n) for n = 0..200
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:1609.06473 [math.CO], 2016.

Cf. A276852, A278391, A278392, A278394, A278395, A278396, A278398.

frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -4, 4}]; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
seq[22] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

seq(n)={my(v=vector(n), m=sum(i=-4, 4, x^i), p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018

A278395 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}.

Original entry on oeis.org

1, 3, 12, 60, 311, 1674, 9173, 51002, 286384, 1620776, 9228724, 52810792, 303447096, 1749612736, 10117583749, 58656027314, 340806249367, 1984018271850, 11569932938192, 67574451148408, 395214184047366, 2314315680481252, 13567587349336459, 79621279809031310

Offset: 0

David Nguyen, Nov 20 2016

By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.

Andrew Howroyd, Table of n, a(n) for n = 0..200
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:1609.06473 [math.CO], 2016.

Cf. A276852, A278391, A278392, A278393, A278394, A278396, A278398.

frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}] - 1; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
seq[24] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

seq(n)={my(v=vector(n), m=sum(i=-3, 3, x^i)-1, p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018

A278416 Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-4,-3,-2,-1,1,2,3,4}.

Original entry on oeis.org

1, 4, 26, 174, 1231, 8899, 65492, 487646, 3664123, 27723979, 210946444, 1612394958, 12371547879, 95230159650, 735060394986, 5687343753535, 44096482961189, 342530654187820, 2665058975987628, 20765913987073659, 162019898098364055, 1265622208055843635

Offset: 0

Michael Wallner, Nov 21 2016

Andrew Howroyd, Table of n, a(n) for n = 0..200
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:1609.06473 [math.CO], 2016.

Cf. A001405, A047002, A278398, A278396, A205337.

seq[n_] := Module[{v = Table[1, n], m = Sum[ x^i, {i, -4, 4}] - 1, p = 1}, For[i = 2, i <= n, i++, p = Expand[p*m]; p = p - Select[p, Exponent[#, x] < 0&]; v[[i]] = (p /. x -> 1)]; v];
seq[25] (* Jean-François Alcover, Jul 11 2018, after Andrew Howroyd *)

seq(n)={my(v=vector(n), m=sum(i=-4, 4, x^i)-1, p=1); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018

cp's OEIS Frontend

A278391 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-2,-1,0,1,2}.

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A278394 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-2,-1,1,2}.

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A278392 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,0,1,2,3}.

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A278393 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-4,-3,-2,-1,0,1,2,3,4}.

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A278395 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}.

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A278416 Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-4,-3,-2,-1,1,2,3,4}.

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