A104455 -id:A104455 - cp's OEIS frontend

A349534 G.f. A(x) satisfies A(x) = 1 / ((1 - 3 * x) * (1 - x * A(x)^2)).

1, 4, 21, 138, 1063, 9075, 82770, 789204, 7766721, 78267306, 803447526, 8371413999, 88300495746, 941004684748, 10116276976218, 109578418285452, 1194764348642313, 13102287157827918, 144422108994233625, 1599198859915070640, 17780781456147340764

Offset: 0

Ilya Gutkovskiy, Nov 21 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..931

Cf. A001764, A104455, A199475, A349254, A349533, A349535.

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

a(n) = 3^n + 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+k,2*k) * binomial(3*k,k) * 3^(n-k) / (2*k+1).
a(n) = 3^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^2/2^4), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021
a(n) ~ sqrt(5/Pi) * 3^(n-1) * 4^n / n^(3/2). - Vaclav Kotesovec, Nov 22 2021

A349603 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k^(n-k).

Original entry on oeis.org

1, 1, 4, 20, 126, 937, 7938, 74909, 775022, 8688827, 104608026, 1342844846, 18273663268, 262347913479, 3957524475778, 62511713866200, 1030842278673510, 17700339693712731, 315740112103311666, 5839137279831300536, 111749137533005481700, 2209538389126578658875

Offset: 0

Vaclav Kotesovec, Nov 23 2021

nonn

Cf. A000248, A007317, A064613, A104455, A104498, A154623, A292632.

Join[{1}, Table[Sum[Binomial[n, j]*CatalanNumber[j]*j^(n-j), {j, 0, n}], {n, 1, 25}]]

a(n) = sum(k=0, n, binomial(n,k)*(binomial(2*k,k)/(k+1))*k^(n-k)); \\ Michel Marcus, Nov 23 2021

A257389 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 6, 6, 17, 21, 54, 74, 183, 272, 644, 1025, 2342, 3928, 8734, 15264, 33227, 59989, 128484, 238008, 503563, 952038, 1995955, 3835381, 7987092, 15548654, 32223061, 63388488, 130918071, 259724317, 535168956, 1069025128

Offset: 0

José Luis Ramírez Ramírez, Apr 21 2015

nonn

			For n=6 we have 6 paths: UDUDUD, H3H3, UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).

Robert Israel, Table of n, a(n) for n = 0..3087

Cf. A000108, A090344, A086622, A257178, A007317, A064613, A104455, A104498.

f:= gfun:-rectoproc({(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6), a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 2},a(n),remember):
map(f, [$0..100]); # Robert Israel, Nov 04 2019

a(n):=sum(((-1)^(n-3*k)+1)*((binomial((n-k)/2,k) )*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2))),k,0,(n)/3); /* Vladimir Kruchinin, Apr 02 2016 */

G.f.: (1-x^3-sqrt((1-x^3)*(1-4*x^2-x^3)))/(2*x^2*(1-x^3)).
a(n) = Sum_{k=0..n/3}(((-1)^(n-3*k)+1)*(binomial((n-k)/2,k)*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2)))). - Vladimir Kruchinin, Apr 02 2016
(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6) = 0. - Robert Israel, Nov 04 2019

A271025 A(n, k) is the n-th binomial transform of the Catalan sequence (A000108) evaluated at k. Array read by descending antidiagonals for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 14, 15, 10, 4, 1, 42, 51, 37, 17, 5, 1, 132, 188, 150, 77, 26, 6, 1, 429, 731, 654, 371, 141, 37, 7, 1, 1430, 2950, 3012, 1890, 798, 235, 50, 8, 1, 4862, 12235, 14445, 10095, 4706, 1539, 365, 65, 9, 1, 16796, 51822, 71398, 56040, 28820, 10392, 2726, 537, 82, 10, 1

Offset: 0

John M. Campbell, Mar 28 2016

Interestingly, the determinant of the n X n array of entries of the form A(i,j) is equal to the (n-1)-th superfactorial number (see A000178).
As indicated in A104455, the k-th binomial transform of A000108 will have:
   o.g.f.: (1-sqrt((1-(k+4)*x)/(1-k*x)))/(2*x),
   e.g.f.: exp((k+2)*x)*(BesselI(0,2x) - BesselI(1,2x)) and
   a(n) = Sum_{i=0..n} binomial(n, i)*CatalanNumber(i)*k^(n-i).
The columns of this array are polynomial integer sequences. The successive polynomials corresponding to the columns of this array are: p0(n) = 1, p1(n) = n + 1, p2(n) = n^2 + 2n + 2, p3(n) = n^3 + 3*n^2 + 6*n + 5, p4(n) = n^4 + 4*n^3 + 12*n^2 + 20*n + 14, and so forth. The coefficients of these successive polynomials form a number triangle, which is given by A098474.

			The array given by integers of the form A(n,k) is illustrated below:
[0] 1, 1,  2,   5,    14,    42,     132,     429,      1430, ...
[1] 1, 2,  5,   15,   51,    188,    731,     2950,     12235, ...
[2] 1, 3,  10,  37,   150,   654,    3012,    14445,    71398, ...
[3] 1, 4,  17,  77,   371,   1890,   10095,   56040,    320795, ...
[4] 1, 5,  26,  141,  798,   4706,   28820,   182461,   1188406, ...
[5] 1, 6,  37,  235,  1539,  10392,  72267,   516474,   3783115, ...
[6] 1, 7,  50,  365,  2726,  20838,  162996,  1303485,  10642310, ...
[7] 1, 8,  65,  537,  4515,  38654,  337007,  2991340,  27013723, ...
[8] 1, 9,  82,  757,  7086,  67290,  648420,  6340365,  62893270, ...
[9] 1, 10, 101, 1031, 10643, 111156, 1174875, 12568686, 136080971, ...
Seen as a triangle:
                          1
                         1, 1
                       2, 2, 1
                      5, 5, 3, 1
                   14, 15, 10, 4, 1
                 42, 51, 37, 17, 5, 1
             132, 188, 150, 77, 26, 6, 1
          429, 731, 654, 371, 141, 37, 7, 1
      1430, 2950, 3012, 1890, 798, 235, 50, 8, 1

Cf. A098474, A000178. Rows 0-5: A000108, A007317, A064613, A104455, A104498, A154623. Columns 0-3: A000012, A000027, A002522, A005491.

A := (n, k) -> (2/Pi)*int((k+4*x^2)^(n-k)*sqrt(1 - x^2), x=-1..1):
for n from 0 to 9 do seq(A(n,k), k=0..n) od; # Peter Luschny, Jan 27 2020

A000108[n_]:= Binomial[2*n,n]/(n+1) ;
T[i_,j_]: Sum[Binomial(j,k)*A000108(k)*i^(j-k), {k,0,j}] ;
A[0, k_] := CatalanNumber[k]; A[n_, k_] := n^k*Hypergeometric2F1[1/2, -k, 2, -4/n];
Table[A[n, k], {n, 0, 6}, {k, 0, 8}] (* Peter Luschny, Jan 27 2020 *)

def A000108(n): return binomial(2*n,n)/(n+1) ;
def T(i,j): return sum(binomial(j,k)*A000108(k)*i^(j-k) for k in range(j+1))

A(0,j) = A000108(j).
A(i,j) = Sum_{k=0..j} binomial(j,k)*A(i-1,k) for i >= 1.
A(i,j) = Sum_{k=0..j} binomial(j,k)*A000108(k)*i^(j-k).
From Peter Luschny, Jan 27 2020: (Start)
A(n,k) = n^k*hypergeom([1/2, -k], [2], -4/n) for n >= 1.
A(n,k) = (2/Pi)*Integral_{x=-1..1}(k + 4*x^2)^(n - k)*sqrt(1 - x^2). (End)

A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level.

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 9, 12, 26, 48, 90, 172, 348, 664, 1349, 2680, 5438, 10976, 22510, 45900, 94700, 195032, 404442, 838824, 1748308, 3646368, 7632628, 15994232, 33606168, 70699504, 149050669, 314625264, 665280246, 1408436672, 2986069782, 6337988876

Offset: 0

José Luis Ramírez Ramírez, Apr 27 2015

nonn

			For n=6 we have 9 paths: UDUDUD, H3H3 (4 options), UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A090344, A086622, A257178, A257388, A007317, A064613, A104455, A104498, A257389.

CoefficientList[Series[(1-2*x^3-Sqrt[(1-2x^3)*(1-4*x^2-2*x^3)])/(2*x^2*(1-2*x^3)), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 28 2015 *)

a(n):=sum((binomial(2*m,m)/(m+1)*(if mod(n+m,3)=0 then 2^((n-2*m)/3)* binomial((m+n)/3,m) else 0)),m,0,n); /* Vladimir Kruchinin, Mar 07 2016 */

seq(n)={Vec((1-2*x^3-sqrt((1-2*x^3)*(1-4*x^2-2*x^3) + O(x^(3+n))))/(2*x^2*(1-2*x^3)))} \\ Andrew Howroyd, May 01 2020

G.f.: (1-2*x^3-sqrt((1-2x^3)*(1-4*x^2-2*x^3)))/(2*x^2*(1-2*x^3)).

Conjecture: (n+2)*a(n) +(n+1)*a(n-1) +(n+4)*a(n-2) +4*(-2*n+3)*a(n-3) +4*(-6*n+17)*a(n-4) +4*(-3*n+10)*a(n-5) +4*(3*n-11)*a(n-6) +4*(11*n-50)*a(n-7) +20*(n-6)*a(n-8)=0. - R. J. Mathar, Jun 07 2016

Terms a(31) and beyond from Andrew Howroyd, May 01 2020

A346763 G.f. A(x) satisfies: A(x) = 1 / (1 - 3x) + x (1 - 3x) A(x)^3.

Original entry on oeis.org

1, 4, 18, 93, 550, 3636, 26079, 197931, 1562382, 12685116, 105187512, 886700898, 7574331987, 65413265014, 570155069547, 5008957733472, 44306834969838, 394269180748272, 3527034255411864, 31700659283908242, 286124960854479888, 2592334353741781752, 23567790327842864046

Offset: 0

Ilya Gutkovskiy, Aug 02 2021

nonn

Third binomial transform of A001764.

Cf. A001764, A104455, A188687, A346762.

nmax = 22; A[] = 0; Do[A[x] = 1/(1 - 3 x) + x (1 - 3 x) A[x]^3 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
Table[3^n HypergeometricPFQ[{1/3, 2/3, -n}, {1, 3/2}, -9/4], {n, 0, 22}]

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

a(n) ~ 3^(n - 5/2) * 13^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 2^(2*(n+1))). - Vaclav Kotesovec, Nov 26 2021

cp's OEIS Frontend

A349534 G.f. A(x) satisfies A(x) = 1 / ((1 - 3 * x) * (1 - x * A(x)^2)).

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A349603 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k^(n-k).

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A257389 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level.

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A271025 A(n, k) is the n-th binomial transform of the Catalan sequence (A000108) evaluated at k. Array read by descending antidiagonals for n >= 0 and k >= 0.

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A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level.

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A346763 G.f. A(x) satisfies: A(x) = 1 / (1 - 3*x) + x * (1 - 3*x) * A(x)^3.

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A346763 G.f. A(x) satisfies: A(x) = 1 / (1 - 3x) + x (1 - 3x) A(x)^3.