A073155 -id:A073155 - cp's OEIS frontend

A073153 Triangle of numbers relating two sequences A073155 and A073156.

1, 1, 2, 4, 5, 9, 14, 18, 22, 36, 56, 70, 86, 100, 156, 237, 293, 349, 405, 461, 698, 1046, 1283, 1507, 1703, 1927, 2164, 3210, 4762, 5808, 6756, 7540, 8324, 9272, 10318, 15080, 22198, 26960, 31144, 34462, 37598, 40916, 45100, 49862, 72060, 105430, 127628

Offset: 0

Paul D. Hanna, Jul 29 2002

			T(4,0) = T(3,3) + 2*T(2,2) + T(1,1) = 2 + 2*9 + 36 = 56.
T(5,2) = A073155(0)*A073155(5) + A073155(1)*A073155(4) + A073155(2)*A073155(3) = 1*237 + 1*56 + 4*14 = 349.
Triangle begins:
1;
1, 2;
4, 5, 9;
14, 18, 22, 36;
56, 70, 86, 100, 156;
237, 293, 349, 405, 461, 698;
1046, 1283, 1507, 1703, 1927, 2164, 3210; ...

Cf. A073154, A073155, A073156, A073157, A073153.

Triangle {T(n, k), n >= 0, 0<=k<=n} defined by:  T(0, 0)=1,
T(n, 0) = A073155(n),  T(n, n) = A073156(n),
T(n, 0) = T(n-1, n-1) + 2*T(n-2, n-2) + T(n-3, n-3),
T(n, k) = Sum_{j=0..k} A073155(j) * A073155(n-j).

Formulas simplified by Paul D. Hanna, Nov 26 2012

a(31) onward corrected by Sean A. Irvine, Nov 18 2024

A073154 Triangle of numbers relating two sequences (A073157 and A073155).

Original entry on oeis.org

1, 2, 4, 5, 9, 14, 18, 28, 38, 56, 70, 106, 131, 167, 237, 293, 433, 523, 613, 753, 1046, 1283, 1869, 2219, 2543, 2893, 3479, 4762, 5808, 8374, 9839, 11099, 12359, 13824, 16390, 22198

Offset: 0

Paul D. Hanna, Jul 29 2002

			a(4,0)=a(3,3)+a(2,2)=56+14=70.
a(5,2)=A073157(0)*A073157(5)+A073157(1)*A073157(4)+A073157(2)*A073157(3)= 1*293+2*70+5*18=523.
Rows:
  {1};
  {2,4};
  {5,9,14};
  {18,28,38,56};
  {70,106,131,167,237};
  {293,433,523,613,753,1046};
  {1283,1869,2219,2543,2893,3479,4762};
  ...

Cf. A073153, A073155, A073156, A073157, A073152.

Triangle {a(n, k), n >= 0, 0<=k<=n} defined by: a(0, 0)=1, a(n, 0)=A073157(n), a(n, n)=A073155(n+1), a(n, 0)=a(n-1, n-1) + a(n-2, n-2), a(n, k)=sum{j=0..k} A073157(j) * A073157(n-j).

G.f.: Sum_{n>=0, 0<=k<=n} a(n, k) x^n y^k = A(x*y)(A(x) - y A(x*y))/(1 - y) where A(x) = (1 - (1 - 4 x (1 + x)^2)^(1/2))/(2 x (1 + x)) is the o.g.f. for A073157. - David Callan, Aug 16 2006

A073157 Number of Schroeder n-paths containing no FFs.

Original entry on oeis.org

1, 2, 5, 18, 70, 293, 1283, 5808, 26960, 127628, 613814, 2990681, 14730713, 73229291, 366936231, 1851352820, 9397497758, 47957377934, 245903408244, 1266266092112, 6545667052320, 33954266444498, 176689391245146

Offset: 0

Paul D. Hanna, Jul 29 2002

Number of Schroeder n-paths containing no FFs. A Schroeder n-path (A006318) consists of steps U=(1,1),F=(2,0),D=(1,-1) starting at (0,0), ending at (2n,0), and never going below the x-axis. Example: a(2)=5 counts UFD, UUDD, UDF, FUD, UDUD. - David Callan, Aug 23 2011

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ...

Muniru A Asiru, Table of n, a(n) for n = 0..300

Leftmost column of triangle A073154 (was previous name).

Cf. A000108, A073155, A073156, A073153, A143330, A198953, A215715, A216359, A364335, A364371.

List([0..25],n->Sum([0..n],i->Binomial(2*i+1,i)*Binomial(2*i+1,n-i)/(2*i+1))); # Muniru A Asiru, Oct 11 2018

a:=n->add(binomial(2*i+1,i)*binomial(2*i+1,n-i)/(2*i+1),i=0..n): seq(a(n),n=0..25); # Muniru A Asiru, Oct 11 2018

Table[Sum[Binomial[2*i + 1, i]*Binomial[2*i + 1, n - i]/(2*i + 1), {i, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 11 2018 *)

a(n):=sum((sum((binomial(2*k+2,j-k)*binomial(2*k,k)/(k+1)),k,0,j))*(-1)^(n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */

{a(n)=local(A=1); for(i=0,n-1,A=(1+x)*(1+x*(A+x*O(x^n))^2));polcoeff(A,n)} /* Paul D. Hanna, Mar 03 2008 */

A073155(n+1) = Sum_{k=0..n} a(k)*a(n-k), that is, convolution yields sequence A073155 minus the 0th term.
G.f.: A(x) = (1 - sqrt(1 - 4*x*(1+x)^2))/(2*x*(1+x)) satisfies A(x) = (1+x)*(1 + x*A(x)^2);
G.f.: A(x) = (1+x)*C(x*(1+x)^2) where C(x) is the Catalan g.f. of A000108. - Paul D. Hanna, Mar 03 2008
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*k+2,j-k)*C(k))))*(-1)^(n-j)), where C(k) = A000108(k). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{i=0..n} C(2*i+1,i)*C(2*i+1,n-i)/(2*i+1). - Vladimir Kruchinin, Oct 11 2018
Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + 6*(2*n - 3)*a(n-2) + 6*(2*n - 5)*a(n-3) + 2*(2*n - 7)*a(n-4). - Vaclav Kotesovec, Oct 11 2018
From Peter Bala, Aug 25 2024: (Start)
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 + ... is the g.f. of A198953.
(1/x) * series_reversion(x*A(-x)) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + ... = G(x) + x, where G(x) = (1 - x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x) is the g.f. of A143330. (End)
Define a sequence operator R: {u(n): n >= 0} -> {v(n): n >= 0} by Sum_{n >= 0} v(n)*x^n = (1/x) * series_reversion(x/Sum_{n >= 0} u(n)*x^n). Then R({a(n)}) = A198953, R^2({a(n)}) = A215715 and R^3({a(n)}) = A364335. Cf. A216359. - Peter Bala, Sep 13 2024

More terms from Paul D. Hanna, Mar 03 2008

New name using a comment from David Callan, Peter Luschny, Oct 14 2018

A366221 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^2A(x)^3.

Original entry on oeis.org

1, 1, 5, 25, 145, 905, 5941, 40433, 282721, 2018897, 14661349, 107945993, 803922289, 6045458905, 45840518933, 350100674785, 2690717983169, 20794719218593, 161502488175557, 1259855507859193, 9867012143508305, 77554946281194793, 611575725258403061

Offset: 0

Seiichi Manyama, Oct 04 2023

nonn

Cf. A364475, A366200, A366222.
Cf. A002478, A073155.
Cf. A366176, A366434.

nmax = 22; A[_] = 1;
Do[A[x_] = 1 + x*(1 + x)^2*A[x]^3 + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

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

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

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366434.

A360076 a(n) = Sum_{k=0..n} binomial(3k,n-k) Catalan(k).

Original entry on oeis.org

1, 1, 5, 20, 90, 430, 2136, 10937, 57307, 305822, 1656482, 9083432, 50328114, 281324294, 1584578746, 8984740485, 51242962251, 293772468164, 1691974930584, 9785378133297, 56805049768157, 330880419984832, 1933299689139364, 11328101469158554

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Cf. A052709, A073155, A360082, A360083.

Cf. A000108, A099234.

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

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^3)))

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

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

A360082 a(n) = Sum_{k=0..n} binomial(4k,n-k) Catalan(k).

Original entry on oeis.org

1, 1, 6, 27, 134, 709, 3892, 22004, 127250, 749230, 4476386, 27071344, 165398868, 1019405720, 6330482488, 39571612357, 248796862550, 1572300095758, 9981970108384, 63633339713190, 407162295120570, 2614059813642256, 16834457481559076

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Cf. A052709, A073155, A360076, A360083.

Cf. A000108, A099235.

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

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^4)))

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

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

A366200 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^3A(x)^3.

Original entry on oeis.org

1, 1, 6, 33, 209, 1425, 10206, 75751, 577494, 4495368, 35582439, 285524184, 2317387098, 18990744137, 156918815760, 1305927563487, 10936673012579, 92098612059051, 779391530714589, 6624730079900931, 56532669993156696, 484156547579505717, 4159926573597719575

Offset: 0

Seiichi Manyama, Oct 04 2023

nonn

Cf. A073155, A366216.

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

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

A073156 Main diagonal sequence of triangle A073153.

Original entry on oeis.org

1, 2, 9, 36, 156, 698, 3210, 15080, 72060, 349184, 1711869, 8475494, 42318018, 212843826, 1077391794, 5484472880, 28058940086, 144195777552, 744017466318, 3852968380624, 20019113126120, 104329129258596, 545214946753377

Offset: 0

Paul D. Hanna, Jul 29 2002

Cf. A073153, A073155, A073157.

a(n, r=2, s=2, t=2, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 07 2024

Convolution of sequence A073155: a(n) = Sum_{k=0..n} A073155(k) * A073155(n-k).
G.f.: 1/4*(1-(1-4*x*(1+x)^2)^(1/2))^2/x^2/(1+x)^4. - Vladeta Jovovic, Oct 10 2003
From Seiichi Manyama, Dec 07 2024: (Start)
G.f. A(x) satisfies A(x) = ( 1 + x * (1 + x)^2 * A(x) )^2.
a(n) = Sum_{k=0..n} binomial(2*k+2,k) * binomial(2*k,n-k)/(k+1). (End)

More terms from Vladeta Jovovic, Oct 10 2003

A360083 a(n) = Sum_{k=0..n} binomial(5k,n-k) Catalan(k).

Original entry on oeis.org

1, 1, 7, 35, 189, 1092, 6538, 40278, 253730, 1626858, 10582616, 69669273, 463319257, 3107941405, 21004392887, 142882885210, 977562617826, 6722361860888, 46438235933700, 322111000796428, 2242538435656450, 15665017062799230, 109761527468995102

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A052709, A073155, A360076, A360082.

Cf. A000108, A360090.

Table[Sum[Binomial[5k,n-k]CatalanNumber[k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 13 2025 *)

a(n) = sum(k=0, n, binomial(5*k, n-k)*binomial(2*k, k)/(k+1));

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^5)))

G.f. A(x) satisfies A(x) = 1/(1 - x * (1+x)^5 * A(x)).

G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^5 )).

A366216 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^4A(x)^4.

Original entry on oeis.org

1, 1, 8, 60, 520, 4886, 48384, 497460, 5259872, 56834345, 624819148, 6966612604, 78592083420, 895432704860, 10288759392156, 119089472755860, 1387274092156508, 16251727492295812, 191342076640423136, 2262894045516407118, 26869820052175649836

Offset: 0

Seiichi Manyama, Oct 04 2023

nonn

Cf. A073155, A366200.

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

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

cp's OEIS Frontend

A073153 Triangle of numbers relating two sequences A073155 and A073156.

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A073154 Triangle of numbers relating two sequences (A073157 and A073155).

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A073157 Number of Schroeder n-paths containing no FFs.

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A366221 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^2*A(x)^3.

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A360076 a(n) = Sum_{k=0..n} binomial(3*k,n-k) * Catalan(k).

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A360082 a(n) = Sum_{k=0..n} binomial(4*k,n-k) * Catalan(k).

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A366200 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^3*A(x)^3.

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A073156 Main diagonal sequence of triangle A073153.

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A360083 a(n) = Sum_{k=0..n} binomial(5*k,n-k) * Catalan(k).

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A366221 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^2A(x)^3.

A360076 a(n) = Sum_{k=0..n} binomial(3k,n-k) Catalan(k).

A360082 a(n) = Sum_{k=0..n} binomial(4k,n-k) Catalan(k).

A366200 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^3A(x)^3.

A360083 a(n) = Sum_{k=0..n} binomial(5k,n-k) Catalan(k).

A366216 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^4A(x)^4.