A300387 -id:A300387 - cp's OEIS frontend

A300386 The number of paths of length 7n from the origin to the line y = 2x/5 with unit East and North steps that stay below the line or touch it.

1, 3, 76, 2803, 121637, 5782513, 291437249, 15297882929, 827402061954, 45790180469312, 2580588279994441, 147592910517101281, 8544927937132306600, 499811636639428519226, 29491983283370728013309, 1753398440591481772556798, 104933899400256659634374549, 6316334518803437568442071134

Offset: 0

Bryan T. Ek, Mar 04 2018

Equivalent to nonnegative walks from (0,0) to (7*n,0) with step set [1,2], [1,-5].

			For n=1, the possible walks are EEEEENN, EEEENEN, EEENEEN.

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, A300387, A300388, A300389, A274052.

terms = 18; f[_] = 0;
Do[f[t_] = f[t]^21 t^3 + 2 f[t]^16 t^2 - f[t]^15 t^2 + 3 f[t]^14 t^2 + f[t]^11 t - f[t]^10 t + 2 f[t]^9 t - 2 f[t]^8 t + 3 f[t]^7 t + 1 + O[t]^terms, {terms}];
CoefficientList[f[t], t] (* Jean-François Alcover, Dec 04 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[Binomial[7*k, 2*k]*x^k/(7*k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2021 *)

G.f. satisfies: f=f^21*t^3+2*f^16*t^2-f^15*t^2+3*f^14*t^2+f^11*t-f^10*t+2*f^9*t-2*f^8*t+3*f^7*t+1.
From Peter Bala, Jan 02 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/7)*binomial(7*n, 2*n)*x^n/n ) - Bizley. Cf. A274052.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/7)*binomial(7*n-7*k, 2*n-2*k)*a(k) for n >= 1. (End)
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 3, 161, 9804, 630401, 41789278, 2824792568, 193553976353, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 11 (checked up to p = 101). - Peter Bala, Sep 14 2021
a(n) ~ c * 7^(7*n) / (n^(3/2) * 2^(2*n) * 5^(5*n)), where c = 0.0538519123304380623474844037127876191519207214308040151922885271364215631... = s*sqrt((3 - 2*s + 2*s^2 - s^3 + s^4 + 6*r*s^7 - 2*r*s^8 + 4*r*s^9 + 3*r^2*s^14) / (63 - 56*s + 72*s^2 - 45*s^3 + 55*s^4 + 273*r*s^7 - 105*r*s^8 + 240*r*s^9 + 210*r^2*s^14)) / (2*sqrt(Pi)), where r = 12500/823543 and s = 1.129379978325... is the root of the equation -16807 + 24010*s - 13720*s^2 + 7350*s^3 - 3500*s^4 + 1250*s^5 = 0. - Vaclav Kotesovec, Sep 16 2021

A300389 The number of paths of length 13n from the origin to the line y = 2x/11 with unit East and North steps that stay below the line or touch it.

Original entry on oeis.org

1, 6, 593, 87143, 15149546, 2891511017, 585739005066, 123655688922720, 26908765569970320, 5993187329634638043, 1359541058523676017369, 313029501692713279534165, 72965556751635426636633639, 17184586991024424745328563477, 4083065013894860643162116395527

Offset: 0

Bryan T. Ek, Mar 04 2018

Equivalent to nonnegative walks from (0,0) to (13*n,0) with step set [1,2], [1,-11].

			For n=1, the possible walks are EEEEEEEEEEENN, EEEEEEEEEENEN, EEEEEEEEENEEN, EEEEEEEENEEEN, EEEEEEEENEEEEN, EEEEEEENEEEEN.

Alois P. Heinz, Table of n, a(n) for n = 0..414
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.

m = 15;
Exp[Sum[(1/13) Binomial[13n, 2n] x^n/n, {n, 1, m}]] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2020, after Peter Bala *)

G.f. satisfies: f = f^78*t^6 + 5*f^67*t^5 - f^66*t^5 + 6*f^65*t^5 + 10*f^56*t^4 - 4*f^55*t^4 + 20*f^54*t^4 - 5*f^53*t^4 + 15*f^52*t^4 + 10*f^45*t^3 - 6*f^44*t^3 + 24*f^43*t^3 - 12*f^42*t^3 + 30*f^41*t^3 - 10*f^40*t^3 + 20*f^39*t^3 + 5*f^34*t^2 - 4*f^33*t^2 + 12*f^32*t^2 - 9*f^31*t^2 + 18*f^30*t^2 - 12*f^29*t^2 + 20*f^28*t^2 - 10*f^27*t^2 + 15*f^26*t^2 + f^23*t - f^22*t + 2*f^21*t - 2*f^20*t + 3*f^19*t - 3*f^18*t + 4*f^17*t - 4*f^16*t + 5*f^15*t - 5*f^14*t + 6*f^13*t + 1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/13)*binomial(13*n, 2*n)*x^n/n ) - Bizley.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/13)*binomial(13*n-13*k, 2*n-2*k)*a(k) for n >= 1. (End)
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 6, 1222, 282993, 69239846, 17468997381, 4494716943847, 1172353182893367, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 5 except for p = 11 and p = 13 (checked up to p = 101). - Peter Bala, Sep 14 2021
a(n) ~ c * 13^(13*n) / (n^(3/2) * 2^(2*n) * 11^(11*n)), where c = 0.0250562444901910770802983936320823301923793538303930752981380507191770309... - Vaclav Kotesovec, Sep 16 2021

A300388 The number of paths of length 11n from the origin to the line y = 2x/9 with unit East and North steps that stay below the line or touch it.

Original entry on oeis.org

1, 5, 345, 35246, 4255288, 563796161, 79264265868, 11612106079203, 1753402118587333, 270965910076404428, 42648418241303137766, 6813002989827352100145, 1101807202785456951146158, 180034116076502209139781574, 29677341363243548521326632028, 4929368173228370040701922315332

Offset: 0

Bryan T. Ek, Mar 04 2018

Equivalent to nonnegative walks from (0,0) to (11*n,0) with step set [1,2], [1,-9].

			For n=1, the walks are EEEEEEEEENN, EEEEEEEENEN, EEEEEEENEEN, EEEEEENEEEN, EEEEENEEEEN.

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, A300389, A274256.

terms = 16; f[_] = 0;
Do[f[t_] = f[t]^55 t^5 + 4 f[t]^46 t^4 - f[t]^45 t^4 + 5 f[t]^44 t^4 + 6 f[t]^37 t^3 - 3 f[t]^36 t^3 + 12 f[t]^35 t^3 - 4 f[t]^34 t^3 + 10 f[t]^33 t^3 + 4 f[t]^28 t^2 - 3 f[t]^27 t^2 + 9 f[t]^26 t^2 - 6 f[t]^25 t^2 + 12 f[t]^24 t^2 - 6 f[t]^23 t^2 + 10 f[t]^22 t^2 + f[t]^19 t - f[t]^18 t + 2 f[t]^17 t - 2 f[t]^16 t + 3 f[t]^15 t - 3 f[t]^14 t + 4 f[t]^13 t - 4 f[t]^12 t + 5 f[t]^11 t + 1 + O[t]^terms, {terms}];
CoefficientList[f[t], t] (* Jean-François Alcover, Dec 04 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[Binomial[11*k, 2*k]*x^k/(11*k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2021 *)

G.f. satisfies: f = f^55*t^5 + 4*f^46*t^4 - f^45*t^4 + 5*f^44*t^4 + 6*f^37*t^3 - 3*f^36*t^3 + 12*f^35*t^3 - 4*f^34*t^3 + 10*f^33*t^3 + 4*f^28*t^2 - 3*f^27*t^2 + 9*f^26*t^2 - 6*f^25*t^2 + 12*f^24*t^2 - 6*f^23*t^2 + 10*f^22*t^2 + f^19*t - f^18*t + 2*f^17*t - 2*f^16*t + 3*f^15*t - 3*f^14*t + 4*f^13*t - 4*f^12*t + 5*f^11*t + 1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/11)*binomial(11*n, 2*n)*x^n/n ) - Bizley. Cf. A274256.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/11)*binomial(11*n-11*k, 2*n-2*k)*a(k) for n >= 1. (End)
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 5, 715, 116213, 19954187, 3532860880, 637870220023, 116749388814357, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 5 except for p = 11 (checked up to p = 101). - Peter Bala, Sep 14 2021
a(n) ~ c * 11^(11*n) / (n^(3/2) * 2^(2*n) * 3^(18*n)), where c = 0.0304820662333129164912550234496338371466905844787974500412037592866845093... - Vaclav Kotesovec, Sep 16 2021

A381758 Expansion of exp( Sum_{k>=1} binomial(9k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 8, 372, 24732, 1925394, 163883548, 14773987638, 1386341339430, 133994232166575, 13248555929274096, 1333732204895318366, 136243562694021684648, 14087033746990654649067, 1471456489458490198994856, 155042502964505871862313879, 16459391575059417875255359878

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381752, A381753, A381757.

Cf. A300387.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(9*k-1, 2*k-1)*x^k/k)))

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(9*k-1,2*k-1) * a(n-k).

G.f.: B(x)^2, where B(x) is the g.f. of A300387.

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.

Original entry on oeis.org

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)

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A300386 The number of paths of length 7*n from the origin to the line y = 2*x/5 with unit East and North steps that stay below the line or touch it.

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A300389 The number of paths of length 13*n from the origin to the line y = 2*x/11 with unit East and North steps that stay below the line or touch it.

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A300388 The number of paths of length 11*n from the origin to the line y = 2*x/9 with unit East and North steps that stay below the line or touch it.

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A381758 Expansion of exp( Sum_{k>=1} binomial(9*k-1,2*k-1) * x^k/k ).

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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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A300386 The number of paths of length 7n from the origin to the line y = 2x/5 with unit East and North steps that stay below the line or touch it.

A300389 The number of paths of length 13n from the origin to the line y = 2x/11 with unit East and North steps that stay below the line or touch it.

A300388 The number of paths of length 11n from the origin to the line y = 2x/9 with unit East and North steps that stay below the line or touch it.

A381758 Expansion of exp( Sum_{k>=1} binomial(9k-1,2k-1) * x^k/k ).

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.