A278394 -id:A278394 - cp's OEIS frontend

A278398 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 {-3,-2,-1,1,2,3}.

1, 3, 15, 75, 400, 2169, 11989, 66985, 377718, 2144290, 12240943, 70193305, 404029950, 2332989921, 13508237399, 78399357623, 455959701700, 2656652705422, 15504203678738, 90614205677898, 530288460288008, 3107012752773125, 18223934202102463, 106996319699099591

Offset: 0

David Nguyen, Nov 20 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. A276902, A276852, A278391, A278394.

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; 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); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018

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}.

Original entry on oeis.org

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

A278396 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,-1,1,2,3,4}.

Original entry on oeis.org

1, 4, 22, 146, 1013, 7269, 53156, 394154, 2951950, 22279439, 169175927, 1290970376, 9891573310, 76050920691, 586426828071, 4533349152056, 35122039919110, 272634162463779, 2119948044144136, 16509519223752380, 128747868290672353, 1005273235488567875

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, A278395, 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}] - 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[22] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

seq(n)={my(v=vector(n), m=sum(i=-4, 4, 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

cp's OEIS Frontend

A278398 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 {-3,-2,-1,1,2,3}.

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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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A278396 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,-1,1,2,3,4}.

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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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