A346626 -id:A346626 - cp's OEIS frontend

A366366 G.f. satisfies A(x) = (1 + x/A(x)^4)/(1 - x).

1, 2, -6, 58, -574, 6402, -75878, 939290, -12000318, 157050178, -2094657926, 28368411194, -389079656446, 5393118559938, -75431624084838, 1063251390845338, -15088643098754942, 215396586102923138, -3091050571516120582, 44566089825496186170

Offset: 0

Seiichi Manyama, Oct 08 2023

sign

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366364, A366365.

Cf. A366359.

a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k-1, k)*binomial(4*k-1, n-k)/(5*k-1));

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k-1,k) * binomial(4*k-1,n-k)/(5*k-1).

A107708 Number of paths from (0,0) to (3n,0) that stay in the first quadrant (x,y >= 0) and where each step is (3,0), (2,1), (1,2), or (1,-1).

Original entry on oeis.org

1, 3, 18, 144, 1323, 13176, 138348, 1507977, 16900650, 193536864, 2254630788, 26635735440, 318350663748, 3842488208997, 46770206742342, 573435609537600, 7075551692662875, 87794803094586336, 1094807464312435344

Offset: 0

Emeric Deutsch, Jun 10 2005

nonn

			a(1)=3 because we have H, uD and Udd, where H=(3,0), u=(2,1), U=(1,2) and D=(1,-1).

G. C. Greubel, Table of n, a(n) for n = 0..880
Emeric Deutsch, Problem 10658, American Math. Monthly, 107, 2000, 368-370.
M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.

Cf. A027307, A106228, A346626.

a:=n->(1/n)*sum(3^j*binomial(n,j)*binomial(n+j,2*n+1-j),j=ceil((n+1)/2)..n): 1,seq(a(n),n=1..22);

Flatten[{1,Table[1/n*Sum[3^j*Binomial[n, j]*Binomial[n+j, 2n+1-j], {j,Floor[(n+1)/2],n}],{n,1,20}]}] (* Vaclav Kotesovec, Mar 17 2014 *)

concat([1], for(n=1,50, print1((1/n)*sum(j=floor((n+1)/2),n, 3^j*binomial(n,j)*binomial(n+j,2*n+1-j)), ", "))) \\ G. C. Greubel, Mar 16 2017

a(n) = (1/n)*Sum(3^j*binomial(n, j)*binomial(n+j, 2n+1-j), j=ceiling((n+1)/2)..n) for n >= 1; a(0)=1.
G.f. = (2/3)w*sin((1/3)*arcsin((36-7z)/2/(3-2z)/w))-1/3, where w=sqrt((3-2z)/z).
Recurrence: 2*n*(2*n+1)*(17*n-25)*a(n) = 4*(238*n^3 - 588*n^2 + 395*n - 72)*a(n-1) - 12*(n-2)*(34*n^2 - 67*n + 21)*a(n-2) + 3*(n-3)*(n-2)*(17*n - 8)*a(n-3). - Vaclav Kotesovec, Mar 17 2014
a(n) ~ (1/204)*sqrt(102)*sqrt((134963 + 21573*sqrt(17))^(1/3) * ((134963 + 21573*sqrt(17))^(2/3) + 2176 + 68*(134963 + 21573*sqrt(17))^(1/3))) / ((134963 + 21573*sqrt(17))^(1/3)*sqrt(Pi)) * 6^(-n) * ((19009 + 153*sqrt(17))^(2/3) + 712 + 28*(19009 + 153*sqrt(17))^(1/3))^n * (19009 + 153*sqrt(17))^(-n/3)*(1/n)^(3/2). - Vaclav Kotesovec, Mar 17 2014
D-finite with recurrence 8*n*(2*n+1)*a(n) +2*(-106*n^2+97*n-18)*a(n-1) +36*(-2*n^2+12*n-15)*a(n-2) +12*(5*n-14)*(n-3)*a(n-3) -9*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
G.f. satisfies A(x) = 1 + x * A(x) * (1 + A(x) + A(x)^2). - Seiichi Manyama, Apr 01 2024

A348793 G.f. A(x) satisfies A(x) = (1 + x * A(x)^3) / (1 - 2 * x).

Original entry on oeis.org

1, 3, 15, 102, 807, 6951, 63240, 597864, 5815167, 57815553, 584919951, 6002197914, 62321630100, 653553174756, 6912106219176, 73642451396160, 789642274208271, 8515008918555573, 92281921130853213, 1004600177464845450, 10980406558088695599, 120454756647900759543

Offset: 0

Ilya Gutkovskiy, Nov 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..937

Cf. A001764, A047891, A346626.

nmax = 21; A[] = 0; Do[A[x] = (1 + x A[x]^3)/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 21}]

a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(n+2*k+1, n)/(n+2*k+1)); \\ Seiichi Manyama, Jul 24 2023

a(0) = 1; a(n) = 2 * a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) ~ sqrt((2 + s^3)/(3*Pi*s*(1 - 2*r))) / (2*n^(3/2)*r^n), where r = (2 + (3*(-2 + sqrt(6))^(1/3))/2^(2/3) - 3/(2*(-2 + sqrt(6)))^(1/3)) / 4 = 0.084819663336750180604484695162155813902734598764355... and s = 1/2 + (-sqrt(2) + sqrt(3))/(2^(5/6)*(-2 + sqrt(6))^(1/3)) + 1/(2*(-2 + sqrt(6)))^(2/3) = 1.8064439323587723772036249693148814564378856424032... - Vaclav Kotesovec, Nov 04 2021
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(n+2*k+1,n) / (n+2*k+1). - Seiichi Manyama, Jul 24 2023

A348912 G.f. A(x) satisfies A(x) = (1 + 2 * x * A(x)^3) / (1 - x).

Original entry on oeis.org

1, 3, 21, 201, 2217, 26535, 335001, 4391553, 59203137, 815580507, 11430639165, 162470033625, 2336381642649, 33930648153615, 496935405133617, 7331179445170689, 108846406625097729, 1625145134034548019, 24385673680861258533, 367546405595389076649, 5561980053932228243529

Offset: 0

Ilya Gutkovskiy, Nov 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..828

Cf. A001764, A103210, A153231, A346626.

nmax = 20; A[] = 0; Do[A[x] = (1 + 2 x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]

a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(n+2*k+1, n)/(n+2*k+1)); \\ Seiichi Manyama, Jul 24 2023

a(0) = 1; a(n) = a(n-1) + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) ~ sqrt(-50 + 30*sqrt(3) + (22 - 12*sqrt(3))*(2*(sqrt(3) - 1))^(1/3) + (2*(sqrt(3) - 1))^(2/3)*(-11 + 7*sqrt(3)))/(4*sqrt(3*Pi)*(-1 + sqrt(3))^(3/2) * n^(3/2) * (1 + (3*(-1 + sqrt(3))^(1/3))/2^(2/3) - 3/(2*(-1 + sqrt(3)))^(1/3))^n). - Vaclav Kotesovec, Nov 04 2021
a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n+2*k+1,n) / (n+2*k+1). - Seiichi Manyama, Jul 24 2023

A363380 G.f. satisfies A(x) = 1 + x * A(x)^4 * (1 + A(x)^2).

Original entry on oeis.org

1, 2, 20, 284, 4712, 85392, 1638112, 32699472, 672188768, 14133399744, 302535052160, 6570819330688, 144442463464704, 3207564324825600, 71848240540852224, 1621452789508328704, 36831997860270007808, 841470878382566444032

Offset: 0

Seiichi Manyama, May 29 2023

Cf. A219534, A346626, A363311.

Cf. A217360, A260332, A363304.

a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+2*k+1, n)/(4*n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+2*k+1,n)/(4*n+2*k+1).

A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2r+k,n)/(n+2r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 8, 0, 1, 6, 20, 44, 0, 1, 8, 36, 120, 280, 0, 1, 10, 56, 236, 800, 1936, 0, 1, 12, 80, 400, 1656, 5696, 14128, 0, 1, 14, 108, 620, 2960, 12192, 42416, 107088, 0, 1, 16, 140, 904, 4840, 22592, 92960, 326304, 834912, 0, 1, 18, 176, 1260, 7440, 38352, 176800, 727824, 2572992, 6652608, 0

Offset: 0

Seiichi Manyama, Nov 20 2024

			Square array begins:
   1,     1,     1,     1,      1,      1,      1, ...
   0,     2,     4,     6,      8,     10,     12, ...
   0,     8,    20,    36,     56,     80,    108, ...
   0,    44,   120,   236,    400,    620,    904, ...
   0,   280,   800,  1656,   2960,   4840,   7440, ...
   0,  1936,  5696, 12192,  22592,  38352,  61248, ...
   0, 14128, 42416, 92960, 176800, 308560, 507152, ...

Columns k=0..1 give A000007, A346626.

Cf. A033877, A071949, A378237, A378238, A378239, A378240.

T(n, k, t=1, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A346626.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+2) for n > 0.

A364552 G.f. satisfies A(x) = 1 + xA(x) + x^4A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 21, 39, 78, 169, 373, 808, 1727, 3719, 8153, 18100, 40315, 89770, 200250, 448755, 1010685, 2284295, 5173961, 11740697, 26699780, 60863291, 139045991, 318247190, 729572315, 1675085099, 3851795549, 8869990949, 20453679944, 47223844863

Offset: 0

Seiichi Manyama, Jul 28 2023

Cf. A000108, A071879, A346626.

Cf. A023426, A063019, A127902.

a(n) = sum(k=0, n\4, binomial(n-k, 3*k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(n-k,3*k) * binomial(3*k,k) / (2*k+1).

A366178 G.f. A(x) satisfies A(x) = 1/(1 - x) + x*A(x)^3/(1 - x)^3.

Original entry on oeis.org

1, 2, 10, 67, 502, 4045, 34279, 301232, 2720266, 25091431, 235394601, 2239139980, 21546299491, 209361514219, 2051379996574, 20245794958408, 201079938971546, 2008276118393320, 20157131084034349, 203215717750220949, 2056913539436637829

Offset: 0

Seiichi Manyama, Oct 03 2023

nonn

Partial sums give A366179.

Cf. A199475, A346626, A366177.

a(n) = sum(k=0, n, binomial(n+4*k, n-k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..n} binomial(n+4*k,n-k) * binomial(3*k,k)/(2*k+1).

A371655 G.f. satisfies A(x) = 1 + x * A(x) * (1 + A(x))^2.

Original entry on oeis.org

1, 4, 32, 336, 4032, 52352, 716032, 10161408, 148229120, 2208921600, 33482670080, 514630230016, 8001860567040, 125640146354176, 1989285578473472, 31725578742464512, 509178657425326080, 8217766225008656384, 133287551280741351424, 2171450128344786403328

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..806

Cf. A106228, A107708, A346626.
Cf. A025225, A371658, A371661.
Cf. A100327.

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 4^(n-k)*binomial(n, k)*binomial(2*n-k, n-1-2*k))/n);

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 4^(n-k) * binomial(n,k) * binomial(2*n-k,n-1-2*k) for n > 0.

a(n) = 2^n * A100327(n). - Seiichi Manyama, Dec 26 2024

A349514 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - 3 * x).

Original entry on oeis.org

1, 4, 24, 192, 1792, 18240, 196224, 2194176, 25247232, 296979456, 3555010560, 43165900800, 530362220544, 6581594275840, 82373440339968, 1038579580796928, 13179023462498304, 168183976239562752, 2157085003249876992, 27790652486543474688, 359485965093121818624

Offset: 0

Ilya Gutkovskiy, Nov 20 2021

nonn

Cf. A001764, A082298, A346626, A348793, A349515.

nmax = 20; A[] = 0; Do[A[x] = (1 + x A[x]^3)/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n+2*k,3*k) * binomial(3*k,k) * 3^(n-k) / (2*k+1)) \\ Andrew Howroyd, Nov 20 2021

a(0) = 1; a(n) = 3 * a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+2*k,3*k) * binomial(3*k,k) * 3^(n-k) / (2*k+1).
a(n) ~ (3/4*(7 + (3*(69 + 16*sqrt(3)))^(1/3) + (3*(69 - 16*sqrt(3)))^(1/3)))^n / (sqrt((4 - (2 + sqrt(3))^(1/3) - (2 - sqrt(3))^(1/3)) * Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 21 2021

cp's OEIS Frontend

A366366 G.f. satisfies A(x) = (1 + x/A(x)^4)/(1 - x).

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A107708 Number of paths from (0,0) to (3n,0) that stay in the first quadrant (x,y >= 0) and where each step is (3,0), (2,1), (1,2), or (1,-1).

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A348793 G.f. A(x) satisfies A(x) = (1 + x * A(x)^3) / (1 - 2 * x).

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A348912 G.f. A(x) satisfies A(x) = (1 + 2 * x * A(x)^3) / (1 - x).

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A363380 G.f. satisfies A(x) = 1 + x * A(x)^4 * (1 + A(x)^2).

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A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2*r+k,n)/(n+2*r+k) for k > 0.

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A364552 G.f. satisfies A(x) = 1 + x*A(x) + x^4*A(x)^3.

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A366178 G.f. A(x) satisfies A(x) = 1/(1 - x) + x*A(x)^3/(1 - x)^3.

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A371655 G.f. satisfies A(x) = 1 + x * A(x) * (1 + A(x))^2.

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A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2r+k,n)/(n+2r+k) for k > 0.

A364552 G.f. satisfies A(x) = 1 + xA(x) + x^4A(x)^3.