A081671 -id:A081671 - cp's OEIS frontend

A302182 Number of 3D walks of type abc.

1, 1, 5, 12, 62, 200, 1065, 3990, 21714, 89082, 492366, 2147376, 12004740, 54718092, 308559537, 1454116950, 8255788970, 39935276810, 227976044010, 1126178350440, 6457854821340, 32456552441040, 186814834574550, 952569927106980, 5500292590186380, 28391993275117500

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A018224, A126120, A135394, A302181.

from math import comb as binomial
def row(n: int) -> list[int]:
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
for n in range(26): print(row(n)) # Mélika Tebni, Nov 27 2024

From Mélika Tebni, Nov 27 2024: (Start)
a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A018224(n-k).
a(2*n+1) = A135394(n) / (2*n+2).
a(2*n) = A302181(n). (End)

a(13)-a(25) from Mélika Tebni, Nov 27 2024

A302184 Number of 3D walks of type abe.

Original entry on oeis.org

1, 2, 7, 26, 108, 472, 2159, 10194, 49396, 244328, 1229308, 6273896, 32410096, 169181664, 891181607, 4731912082, 25302648644, 136150941064, 736747902236, 4007011320808, 21893702201648, 120125750018656, 661630546993116, 3656966382542984, 20278320788680912, 112782556853239712

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A126120, A138547, A145847, A145867, A150500, A202814.

a := n -> 2*add(binomial(n, k)*binomial(k, k/2)*binomial(2*(n-k), n-k)/(k+2), k = 0..n, 2): seq(a(n), n = 0..25);  # Peter Luschny, Nov 30 2024

from math import comb as binomial
def a(n: int):
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(2*(n-k), n-k) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Nov 30 2024

a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A000984(n-k). - Mélika Tebni, Nov 30 2024

a(12)-a(25) from Mélika Tebni, Nov 30 2024

A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

Original entry on oeis.org

1, 3, 4, 14, 46, 174, 656, 2640, 10790, 45340, 193600, 839820, 3686424, 16353924, 73187456, 330052646, 1498335650, 6841899606, 31404443032, 144814450188, 670552118244, 3116578216310, 14534401932712, 67992210407514, 318969964124256, 1500268062754830

Offset: 0

R. J. Mathar, Nov 05 2021

nonn

Counts a subset of the Lyndon words in A001692. Here there is no requirement of how often the 3rd to 5th letter of the alphabet are in the admitted word, only on the frequency of the 1st and 2nd letter of the alphabet.

Let T(n,k,M) be the number of words of length n drawn from an alphabet of size M where the first k letters of the alphabet appear with the same frequency f in each word. Then T(n,k,M) = Sum_{f=0..floor(n/k)} (M-k)^(n-f*k) * Product_{i=0..k-1} binomial(n-i*f,f) and T(n,2,5) = A026375(n), T(n,3,6) = A294035(n), T(n,2,6) = A081671(n). Removing the words with cycles by the inclusion-exclusion principle by a Mobius Transform gives words of length n of that type without cycles and division through n the Lyndon words of that type. - R. J. Mathar, Nov 07 2021

			Examples for the alphabet {0,1,2,3,4}:
a(0)=1 counts (), the empty word.
a(3)=14 counts (021) (031) (041) (012) (013) (223) (233) (243) (014) (224) (234) (334) (244) (344), words of length 3 where the letters 0 and the 1 occur both either not or once.
a(4)=46 counts (0011) (0221) (0321) (0421) (0231) (0331) (0431) (0241) (0341) (0441) (0212) (0312) (0412) (0122) (0132) (0142) (0213) (0313) (0413) (0123) (2223) (0133) (2233) (2333) (2433) (0143) (2243) (2343) (2443) (0214) (0314) (0414) (0124) (2224) (2324) (0134) (2234) (2334) (3334) (2434) (0144) (2244) (2344) (3344) (2444) (3444).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Lyndon words

Cf. A022553 (alphabet of 2 letters), A290277 (of 3 letters), A060165 (of 4 letters), A026375.

a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, binomial(d,k)*binomial(2*k,k)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023

n*a(n) = Sum_{d|n} mu(d)*A026375(n/d) where mu = A008683.

Terms a(16) and beyond from Andrew Howroyd, Jan 14 2023

A302178 The number of 3D walks of semilength n in a quadrant returning to the origin.

Original entry on oeis.org

1, 4, 40, 570, 9898, 195216, 4209084, 96941130, 2349133930, 59272544760, 1545550116240, 41416083787260, 1135679731004700, 31760915181412800, 903492759037272480, 26086451983000501410, 763124703525758894490, 22585374873810849150600, 675419388009799152812400

Offset: 0

N. J. A. Sloane, Apr 09 2018

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5. The sequence is type aab in Table 3.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

a(n) = Sum_{i=0..n,j=0..n-i} A000108(i) * A000108(j) * A000984_(n-i-j) * (2n)!/((2i)!*(2j)!*(2n-2i-2j)!). - Nachum Dershowitz, Aug 13 2020

a(8)-a(18) from Nachum Dershowitz, Aug 03 2020

Name edited by Nachum Dershowitz, Aug 13 2020

A302179 The number of 3D walks of length n in an octant returning to axis of origin.

Original entry on oeis.org

1, 1, 4, 9, 40, 120, 570, 1995, 9898, 38178, 195216, 805266, 4209084, 18239364, 96941130, 436235085, 2349133930, 10891439130, 59272544760, 281544587610, 1545550116240, 7489973640240, 41416083787260, 204122127237210, 1135679731004700, 5678398655023500, 31760915181412800, 160789633105902300

Offset: 0

N. J. A. Sloane, Apr 09 2018

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5. The sequence is type aac in Table 3.
Mélika Tebni, Fonctions de Bessel et cheminements en 3D, Dec 2024.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

C(n) = binomial(2*n, n)/(n+1); \\ A000108
f(n) = binomial(n, floor(n/2)); \\ A001405
a(n) = sum(i=0, n, if (!(i%2), sum(j=0, n-i, if (!(j%2), C(i/2)*C(j/2)*f(n-i-j)*n!/(i! * j! * (n-i-j)!))))); \\ Michel Marcus, Aug 07 2020

a(n) = Sum_{i=0..n, j=0..n-i, i,j even} A126120(i) * A126120(j) * A001405(n-i-j) * n!/(i! * j! * (n-i-j)!). - Nachum Dershowitz, Aug 06 2020

E.g.f.: (BesselI(1, 2*x)/x)^2*(BesselI(0, 2*x) + BesselI(1, 2*x)). - Mélika Tebni, Jan 06 2025

a(13)-a(27) from Nachum Dershowitz, Aug 04 2020

A302183 Number of 3D n-step walks of type abd.

Original entry on oeis.org

1, 1, 4, 10, 39, 131, 521, 1989, 8149, 33205, 139870, 592120, 2552155, 11079303, 48639722, 214997228, 957817013, 4292316197, 19349957108, 87663905954, 399038606291, 1823961268751, 8369603968599, 38540835938335, 178056111047329, 825079806039121, 3833960405339446

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A001006, A126869, A302184.

from math import comb as binomial
def M(n): return sum(binomial(n, 2*k)*binomial(2*k, k)//(k+1) for k in range(n//2+1)) # Motzkin numbers
def a(n):
    return sum(binomial(n, k)*binomial(k, k//2)*((k+1) %2)*M(n-k) for k in range(n+1))
print([a(n) for n in range(27)]) # Mélika Tebni, Dec 03 2024

From Mélika Tebni, Dec 03 2024: (Start)
a(n) = Sum_{k=0..n} binomial(n, k)*A126869(k)*A001006(n-k).
Inverse binomial transform of A302184. (End)

a(13)-a(26) from Mélika Tebni, Dec 03 2024

A302185 Number of 3D n-step walks of type acc.

Original entry on oeis.org

1, 2, 7, 24, 98, 400, 1785, 7980, 37674, 178164, 874146, 4294752, 21667932, 109436184, 563910633, 2908233900, 15235550330, 79870553620, 424021948350, 2252356700880, 12088746573540, 64913104882080, 351594254659830, 1905139854213960, 10399223643879420, 56783986550235000

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A000918, A001405, A005558, A005566, A135394, A302182.

b:= n-> binomial(n, floor(n/2))*binomial(n+1, floor((n+1)/2)):
C:= n-> binomial(2*n, n)/(n+1):
a:= n-> add(binomial(n, 2*k)*C(k)*b(n-2*k), k=0..n/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 06 2024
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 24][n+1],
      (8*(14*n^4+85*n^3+190*n^2+188*n+63)*a(n-1)+4*(n-1)*
      (80*n^4+418*n^3+676*n^2+269*n-108)*a(n-2)-96*(n-1)*(n-2)*
      (10*n^2+31*n+27)*a(n-3)-144*(n-1)*(n-2)*(n-3)*(8*n^2+33*n+36)*
       a(n-4))/((n+4)*(n+3)*(n+2)*(8*n^2+17*n+11)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 06 2024

b[n_] := Binomial[n, Floor[n/2]]*Binomial[n+1, Floor[(n+1)/2]];
c[n_] := Binomial[2*n, n]/(n+1);
a[n_] := Sum[Binomial[n, 2*k]*c[k]*b[n - 2*k], {k, 0, n/2}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 28 2025, after Alois P. Heinz *)

from math import comb as binomial
def C(n): return (binomial(2*n, n)//(n+1)) # Catalan numbers
def a(n):
    return sum(binomial(n, k)*C((k+1)//2)*C(k//2)*(2*(k//2)+1)*binomial(n-k, (n-k)//2) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Dec 06 2024

From Mélika Tebni, Dec 06 2024: (Start)
E.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*BesselI(1, 2*x) / x.
a(n) = Sum_{k=0..n} binomial(n, k)*A005558(k)*A001405(n-k).
a(2*n+1) = 2*A302182(2*n+1) = A135394(n) / (n+1).
For n > 0, a(A000918(n)) is odd. (End)

a(13)-a(25) from Mélika Tebni, Dec 06 2024

A302186 Number of 3D walks of type ace.

Original entry on oeis.org

1, 3, 11, 44, 188, 842, 3911, 18692, 91412, 455540, 2306028, 11829424, 61375408, 321583108, 1699500055, 9049714852, 48513809796, 261638920412, 1418673379052, 7730011715760, 42305916178288, 232475082183544, 1282208011668988, 7096065370945168, 39394821683770960, 219341739839760912

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867 (number of 3D walks of type acd), A150500, A202814.

from math import comb as binomial
def C(n): return (binomial(2*n, n)//(n+1)) # Catalan numbers
def row(n: int) -> list[int]:
     return sum(binomial(n, k)*sum(binomial(k, j)*C((j+1)//2)*C(j//2)*(2*(j//2)+1) for j in range(k+1)) for k in range(n+1))
for n in range(26): print(row(n)) # Mélika Tebni, Nov 29 2024

Binomial transform of A145847. - Mélika Tebni, Nov 29 2024

a(12)-a(25) from Mélika Tebni, Nov 29 2024

A302187 Number of 3D walks of type bcc.

Original entry on oeis.org

1, 2, 8, 30, 138, 620, 3060, 14910, 76650, 390852, 2063376, 10832052, 58264668, 312123240, 1702423008, 9256786110, 51036229530, 280696824980, 1560925457520, 8663089672380, 48512836025940, 271229902496280, 1527733861191720, 8593482390429300, 48642125421855420, 275014629509319000

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A001405, A018224.

from math import comb as binomial
def a(n):
    return sum(binomial(n, k)*binomial(k, k//2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Nov 25 2024

a(n) = Sum_{k=0..n} binomial(n, k)*A001405(k)*A018224(n-k). - Mélika Tebni, Nov 25 2024

a(12)-a(25) from Nachum Dershowitz, Aug 03 2020

A302188 Number of 3D walks of type bce.

Original entry on oeis.org

1, 3, 12, 53, 252, 1252, 6416, 33609, 178996, 965660, 5263728, 28936404, 160204336, 892313424, 4995832640, 28096475977, 158638993476, 898844200524, 5108695394096, 29117034808980, 166370716319088, 952789631705104, 5467881256289856, 31438798094242244, 181079794531199440, 1044651995141484912

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Binomial transform of A150500 (Number of 3D walks of type bcd). - Mélika Tebni, Nov 28 2024

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

from math import comb as binomial
def a(n):
    return sum(binomial(n, k)*sum(binomial(k, j)*binomial(j, j//2)**2 for j in range(k+1)) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Nov 28 2024

a(12)-a(25) from Mélika Tebni, Nov 28 2024

cp's OEIS Frontend

A302182 Number of 3D walks of type abc.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

Extensions

A302184 Number of 3D walks of type abe.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Python

Formula

Extensions

A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A302178 The number of 3D walks of semilength n in a quadrant returning to the origin.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A302179 The number of 3D walks of length n in an octant returning to axis of origin.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A302183 Number of 3D n-step walks of type abd.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

Extensions

A302185 Number of 3D n-step walks of type acc.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A302186 Number of 3D walks of type ace.

Views