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
- 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.
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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 *)
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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
- 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.
-
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 *)
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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
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
- 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.
-
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 *)
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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
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
- 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.
-
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
- 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.
-
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 *)
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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
Showing 1-5 of 5 results.
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