A060941 -id:A060941 - cp's OEIS frontend

A300390 The number of paths of length 7n from the origin to the line y = 3x/4 with unit east and north steps that stay below the line or touch it.

1, 5, 227, 15090, 1182187, 101527596, 9247179818, 877362665128, 85783306955099, 8582893111512001, 874542924575207352, 90437361732467946334, 9467275300762187682554, 1001309098267187214993056, 106836493655355495755649544, 11485688815900189437990930096, 1242964338344397490958154292155

Offset: 0

Bryan T. Ek, Mar 05 2018

Equivalent to nonnegative walks from (0,0) to (7*n,0) with step set [1,3], [1,-4].

			For n=1, the possible walks are EEEENNN, EEENENN, EENEENN, EEENNEN, EENENEN.

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A001764, A060941, A300386, A300387, A300388, A300389, A300391.

m = 17; f = 0; Do[f = f^35*t^5 - f^31*t^4 + f^30*t^4 - f^29*t^4 + 5*f^28*t^4 - f^25*t^3 + f^24*t^3 + 3*f^23*t^3 - 4*f^22*t^3 + 10*f^21*t^3 + f^19*t^2 - f^18*t^2 + 5*f^17*t^2 + 3*f^16*t^2 - 6*f^15*t^2 + 10*f^14*t^2 + f^13*t - f^12*t + 3*f^10*t + f^9*t - 4*f^8*t + 5*f^7*t + 1 + O[t]^m, {m}]; CoefficientList[f, t] (* Jean-François Alcover, Feb 18 2019 *)

G.f. satisfies: f = f^35*t^5 - f^31*t^4 + f^30*t^4 - f^29*t^4 + 5*f^28*t^4 - f^25*t^3 + f^24*t^3 + 3*f^23*t^3 - 4*f^22*t^3 + 10*f^21*t^3 + f^19*t^2 - f^18*t^2 + 5*f^17*t^2 + 3*f^16*t^2 - 6*f^15*t^2 + 10*f^14*t^2 + f^13*t - f^12*t + 3*f^10*t + f^9*t - 4*f^8*t + 5*f^7*t + 1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/7)*binomial(7*n, 3*n)*x^n/n ) - Bizley.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/7)*binomial(7*n-7*k, 3*n-3*k)*a(k) for n >= 1. (End)

A300391 The number of paths of length 8n from the origin to the line y = 3x/5 with unit east and north steps that stay below the line or touch it.

Original entry on oeis.org

1, 7, 525, 58040, 7574994, 1084532963, 164734116407, 26070940600055, 4252443527211637, 709846349042619913, 120679177855928146859, 20822762876863605793639, 3637213213067542990001936, 641912742432770594132245835, 114287840570892852593437353124, 20502971288127330644273350110698

Offset: 0

Bryan T. Ek, Mar 05 2018

Equivalent to nonnegative walks from (0,0) to (8*n,0) with step set [1,3], [1,-5].

			For n=1, the possible walks are EEEEENNN, EEEENENN, EEEENNEN, EEENEENN, EEENENEN, EENEEENN, EENEENEN.

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A001764, A060941, A300386, A300387, A300388, A300389, A300390.

G.f. f satisfies f = t^7*f^56 - 2*t^6*f^51 + t^6*f^50 - t^6*f^49 + 7*t^6*f^48 + t^5*f^46 - t^5*f^45 - 3*t^5*f^43 + 5*t^5*f^42 - 6*t^5*f^41 + 21*t^5*f^40 - 3*t^4*f^37 - 3*t^4*f^36 + 8*t^4*f^35 + 10*t^4*f^34 - 15*t^4*f^33 + 35*t^4*f^32 - 2*t^3*f^31 + 2*t^3*f^30 - 9*t^3*f^28 + 22*t^3*f^27 + 10*t^3*f^26 - 20*t^3*f^25 + 35*t^3*f^24 + 3*t^2*f^22 + 5*t^2*f^21 - 9*t^2*f^20 + 18*t^2*f^19 + 5*t^2*f^18 - 15*t^2*f^17 + t*(21*t + 1)*f^16 - t*f^15 + 3*t*f^13 - 3*t*f^12 + 5*t*f^11 + t*f^10 - 6*t*f^9 + 7*t*f^8 + 1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/8)*binomial(8*n, 3*n)*x^n/n ) - Bizley.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/8)*binomial(8*n-8*k, 3*n-3*k)*a(k) for n >= 1. (End)

A165997 Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 1, 4, 9, 9, 1, 5, 14, 23, 23, 1, 6, 20, 43, 66, 1, 7, 27, 70, 136, 136, 1, 8, 35, 105, 241, 377, 377, 1, 9, 44, 149, 390, 767, 1144, 1, 10, 54, 203, 593, 1360, 2504, 2504, 1, 11, 65, 268, 861, 2221, 4725, 7229, 7229, 1, 12, 77, 345, 1206

Offset: 0

Gerald McGarvey, Oct 03 2009

There are f(n) = floor((2*n+3)/3) = A004396(n+1) terms in row n.

			Triangle begins:
       k=0   1   2    3     4     5     6      7      8
  n=0:   1
  n=1:   1
  n=2:   1,  1
  n=3:   1,  2,  2
  n=4:   1,  3,  5
  n=5:   1,  4,  9,   9
  n=6:   1,  5, 14,  23,   23
  n=7:   1,  6, 20,  43,   66
  n=8:   1,  7, 27,  70,  136,  136
  n=9:   1,  8, 35, 105,  241,  377,  377
  n=10:  1,  9, 44, 149,  390,  767, 1144
  n=11:  1, 10, 54, 203,  593, 1360, 2504,  2504
  n=12:  1, 11, 65, 268,  861, 2221, 4725,  7229,  7229
  n=13:  1, 12, 77, 345, 1206, 3427, 8152, 15381, 22610
  ...

Cf. A004396 (row lengths).

Cf. A060941 (diagonal T(3*n, 2*n)).

f(n) = floor((2*(n-1)+3)/3); s=14; M=matrix(s,s); for(n=1,s,M[n,1]=1); for(n=2,s,for(k=2,f(n),M[n,k]=M[n,k-1]+M[n-1,k])); for(n=1,s,for(k=1,f(n),print1(M[n,k],", ")))

cp's OEIS Frontend

A300390 The number of paths of length 7n from the origin to the line y = 3x/4 with unit east and north steps that stay below the line or touch it.

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A300391 The number of paths of length 8n from the origin to the line y = 3x/5 with unit east and north steps that stay below the line or touch it.

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A165997 Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.

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cp's OEIS Frontend

A300390 The number of paths of length 7*n from the origin to the line y = 3*x/4 with unit east and north steps that stay below the line or touch it.

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A300391 The number of paths of length 8*n from the origin to the line y = 3*x/5 with unit east and north steps that stay below the line or touch it.

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Formula

A165997 Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.

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A300390 The number of paths of length 7n from the origin to the line y = 3x/4 with unit east and north steps that stay below the line or touch it.

A300391 The number of paths of length 8n from the origin to the line y = 3x/5 with unit east and north steps that stay below the line or touch it.