David Nguyen - cp's OEIS frontend

A354709 Decimal expansion of Sum_{p prime} 3(2p-1)log(p)/(p^3 + p^2 - 3p + 1).

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

Offset: 1

David Nguyen, Jun 03 2022

Also logarithmic derivative of A(s,w) at (0,0), where A(s,w) = Product_{p prime} (1 - (1 - (p*(1 - p^(-1-s))^3)/(-1+p))*(1 - (p*(1 - p^(-1-w))^3)/(-1+p))), with A(0,0) = A256392.

			2.52906617358092992925958712930189459230009223994439976118899256270135780066...

Cf. A256392.

Block[{$MaxExtraPrecision = 1000},
Do[CC = Join[{0},
    Series[(3 (-1 + 2 p))/(1 - 3 p + p^2 + p^3) //. p -> 1/x, {x, 0,
       t}][[3]]];
  Print[N[-Sum[
        CC[[k]]*(PrimeZetaP'[k] + Log[2]/2^k), {k, 1, Length[CC]}] + (
      3 (-1 + 2 p) Log[p])/(1 - 3 p + p^2 + p^3) //. p -> 2, 75]], {t,
    1000, 1500, 100}]]
ratfun = 3*(2*p - 1)/(p^3 + p^2 - 3*p + 1); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 120]], {m, 2000, 20000, 2000}] (* Vaclav Kotesovec, Jun 04 2022 *)

More digits from Vaclav Kotesovec, Jun 04 2022

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}.

Original entry on oeis.org

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

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

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

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

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

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

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

A277923 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 3, 16, 84, 505, 3121, 20180, 133604, 904512, 6224305, 43432093, 306524670, 2184389874, 15695947669, 113595885023, 827299204132, 6058526521135, 44586954104578, 329579179316696, 2445858862779018, 18216235711289695, 136113075865844577, 1020074492384232296

Offset: 0

David Nguyen, Nov 04 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1113
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. A276901, A276852, A276902, A276903, A276904.`

A277922 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 3, 13, 71, 405, 2501, 15923, 104825, 704818, 4827957, 33549389, 235990887, 1676907903, 12019875907, 86804930199, 630999932585, 4613307289260, 33900874009698, 250257489686870, 1854982039556397, 13800559463237465, 103017222722691145, 771348369563479705

Offset: 0

David Nguyen, Nov 04 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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, A276901, A276902, A276903, A276904.

b:= proc(n, y) option remember; `if`(n=0, `if`(y=1, 1, 0), add
     ((h-> `if`(h<1, 0, b(n-1, h)))(y+d), d=[$-4..-1, $1..4]))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Nov 12 2016

b[n_, y_] := b[n, y] = If[n == 0, If[y == 1, 1, 0], Sum[Function[h, If[h < 1, 0, b[n - 1, h]]][y + d], {d, Join[Range[-4, -1], Range[4]]}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 03 2017, after Alois P. Heinz *)

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User: David Nguyen

A354709 Decimal expansion of Sum_{p prime} 3*(2p-1)*log(p)/(p^3 + p^2 - 3p + 1).

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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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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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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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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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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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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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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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Author

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A354709 Decimal expansion of Sum_{p prime} 3(2p-1)log(p)/(p^3 + p^2 - 3p + 1).