A047000 -id:A047000 - cp's OEIS frontend

A047002 T(n,n), array T given by A047000.

1, 1, 2, 7, 23, 83, 299, 1107, 4122, 15523, 58769, 223848, 856085, 3286687, 12656513, 48871469, 189145479, 733547091, 2849962925, 11090427510, 43219527353, 168645172164, 658834266936, 2576566240218, 10086236606187

Offset: 0

Clark Kimberling

nonn

Alternatively, this sequence counts the 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 {-2,-1,1,2}. - David Nguyen, Dec 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 preprint arXiv:1609.06473 [math.CO], 2016.

Cf. A047000, A278398.

seq[n_] := Module[{v = Table[1, n], m = Sum[ x^i, {i, -2, 2}] - 1, p = 1}, For[i = 3, i <= n, i++, p = Expand[p*m]; p = p - Select[p, Exponent[#, x] < 0&]; v[[i]] = ReplaceAll[p, x -> 1]]; v];
seq[25] (* Jean-François Alcover, Jul 11 2018, after Andrew Howroyd *)

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

A047001 Sum{T(i,n-i): i=0,1,...,n}, array T given by A047000.

Original entry on oeis.org

1, 2, 3, 6, 8, 15, 30, 52, 101, 202, 376, 740, 1480, 2840, 5625, 11250, 21928, 43583, 87166, 171420, 341412, 682824, 1350144, 2692536, 5385072, 10684872, 21326481, 42652962, 84825224, 169403773, 338807546, 674852332, 1348273949

Offset: 0

Clark Kimberling

nonn

A047003 T(n,n-1), array T given by A047000.

Original entry on oeis.org

1, 2, 2, 7, 30, 113, 442, 1692, 6542, 25213, 97548, 377617, 1465010, 5690787, 22140830, 86250091, 336416094, 1313641661, 5134950280, 20091508863, 78682553016, 308392187964, 1209658474836, 4748241492783, 18650595931506

Offset: 1

Clark Kimberling

nonn

A047004 T(n,n-2), array T given by A047000.

Original entry on oeis.org

1, 1, 4, 7, 30, 143, 585, 2420, 9690, 38779, 153769, 608925, 2404100, 9484317, 37378622, 147270615, 580094570, 2284987355, 9001081353, 35463025663, 139747015800, 550824207900, 2171675252565, 8564359325319, 33784352403732

Offset: 2

Clark Kimberling

nonn

A047005 T(n,n-3), array T given by A047000.

Original entry on oeis.org

1, 2, 3, 14, 30, 143, 728, 3148, 13566, 56221, 231308, 939090, 3793530, 15237792, 61020524, 243678476, 971345694, 3866131073, 15371516800, 61064462784, 242432019936, 962016777729, 3816117832536, 15133756472226

Offset: 3

Clark Kimberling

nonn

A047006 T(n,n+1), array T given by A047000.

Original entry on oeis.org

1, 2, 5, 16, 53, 186, 665, 2430, 8981, 33556, 126300, 478468, 1821677, 6965726, 26730639, 102895388, 397130997, 1536321264, 5955477230, 23128018490, 89962619148, 350442078972, 1366907765382, 5337995113404, 20868301796085

Offset: 0

Clark Kimberling

nonn

A047007 T(n,n+2), array T given by A047000.

Original entry on oeis.org

1, 3, 9, 30, 103, 366, 1323, 4859, 18033, 67531, 254620, 965592, 3679039, 14074126, 54023919, 207985518, 802774173, 3105514305, 12037590980, 46743091795, 181796906808, 708073498446, 2761428873186, 10782065189898, 42144216934185

Offset: 0

Clark Kimberling

nonn

A047008 T(n,n+3), array T given by A047000.

Original entry on oeis.org

1, 4, 14, 50, 180, 658, 2429, 9052, 33975, 128320, 487124, 1857362, 7108400, 27293280, 105090130, 405643176, 1569193041, 6082113750, 23615073305, 91834287660, 357631419474, 1394521107804, 5444070076494, 21275915138100

Offset: 0

Clark Kimberling

nonn

A047009 T(2n,n), array T given by A047000.

Original entry on oeis.org

1, 2, 4, 14, 60, 286, 1456, 7752, 42636, 240350, 1381380, 8064030, 47682960, 284997384, 1719031840, 10450528048, 63967345068, 393895175646, 2438398706380, 15166284983850, 94730949283740, 593966352738990

Offset: 0

Clark Kimberling

nonn

Also: number of nonseparable planar rooted maps with n edges. - R. J. Mathar, Aug 18 2006

T. R. S. Walsh, A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.

a(n) = 2*A006013(n-1) = 2*(3n-2)!/(n!*(2*n-1)!). - R. J. Mathar, Aug 18 2006

G.f.: 1+2*g where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

A047070 Array T read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=2x unless x=0 or x=h.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 5, 4, 1, 1, 3, 4, 7, 9, 5, 1, 1, 4, 3, 7, 16, 14, 6, 1, 1, 5, 7, 7, 23, 30, 20, 7, 1, 1, 6, 12, 14, 30, 53, 50, 27, 8, 1, 1, 7, 18, 12, 30, 83, 103, 77, 35, 9, 1, 1, 8, 25, 30, 30, 113, 186, 180, 112, 44

Offset: 0

Clark Kimberling

			Diagonals (beginning on row 0): {1}; {1,1}; {1,1,1}; {1,2,2,1}; {1,1,2,3,1}; ...

Cf. A047000 (mirror image).

a(0) = 1 inserted by Sean A. Irvine, May 11 2021

cp's OEIS Frontend

A047002 T(n,n), array T given by A047000.

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A047001 Sum{T(i,n-i): i=0,1,...,n}, array T given by A047000.

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A047003 T(n,n-1), array T given by A047000.

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A047004 T(n,n-2), array T given by A047000.

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A047005 T(n,n-3), array T given by A047000.

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A047006 T(n,n+1), array T given by A047000.

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A047007 T(n,n+2), array T given by A047000.

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A047008 T(n,n+3), array T given by A047000.

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A047009 T(2n,n), array T given by A047000.

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A047070 Array T read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=2x unless x=0 or x=h.

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