A086621 -id:A086621 - cp's OEIS frontend

A086622 G.f. A(x) satisfies: A(x) = 1/(1-2x) + x^2A(x)^2.

1, 2, 5, 12, 30, 76, 197, 520, 1398, 3820, 10594, 29768, 84620, 243000, 704045, 2055760, 6043750, 17875020, 53148310, 158773320, 476311940, 1434313960, 4333867170, 13135533552, 39924668220, 121661345656, 371612931492

Offset: 0

Paul D. Hanna, Jul 24 2003

nonn

a(n) is the number of Motzkin paths of length n having no (1,0)-steps at levels 1,3,5,... and having (1,0)-steps of two colors at levels 2,4,6,... . Example: a(3) = 12 because, denoting U=(1,1), D=(1,-1), and H=(1,0), we have 8 paths of shape HHH, 2 paths of shape HUD, and 2 paths of shape UDH. - Emeric Deutsch, May 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Carles Cardó, Growth and density in free magmas, arXiv:2401.07827 [math.CO], 2024. See p. 16.

Cf. A086620 (table), A086621 (diagonal).

A086622 := proc(n)
    option remember;
    if n < 3 then
        op(n+1,[1,2,5]) ;
    else
        4*(-n-1)*procname(n-1) +4*procname(n-2) +4*(2*n-3)*procname(n-3) ;
        -%/(n+2) ;
    end if;
end proc:
seq(A086622(n),n=0..20) ; # R. J. Mathar, Nov 02 2021

CoefficientList[Series[(-1+2*x+Sqrt[1-4*x+8*x^3])/(2*(-x^2+2*x^3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

Antidiagonal sums of square table A086620.
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k) C(2k,k) 2^(n-2k)/(k+1). - Paul Barry, Nov 13 2004
Hankel transform of a(n) is 1,1,1,....; Hankel transform of a(n+1) is A009531(n+2). - Paul Barry, Nov 06 2007
G.f.: 1/(1-2*x-x^2/(1-x^2/(1-2*x-x^2/(1-x^2/(1-2*x-x^2/..... (continued fraction). - Paul Barry, Dec 21 2008
D-finite with recurrence (n+2)*a(n) +4*(-n-1)*a(n-1) +4*a(n-2) +4*(2*n-3)*a(n-3)=0. - R. J. Mathar, Nov 24 2012
G.f.: (-1+2*x+sqrt(1-4*x+8*x^3))/(2*(-x^2+2*x^3)). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ sqrt(50+22*sqrt(5)) * (sqrt(5)+1)^n / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014. Equivalently, a(n) ~ 5^(1/4) * 2^n * phi^(n + 5/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
a(n) = Sum_{i=0..floor(n/2)}2^(n-2i)*C(i)*binomial(n-i,i), where C(n) is the n-th Catalan number A000108. - José Luis Ramírez Ramírez, Apr 20 2015

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 28, 28, 9, 1, 1, 11, 47, 79, 47, 11, 1, 1, 13, 71, 175, 175, 71, 13, 1, 1, 15, 100, 331, 504, 331, 100, 15, 1, 1, 17, 134, 562, 1196, 1196, 562, 134, 17, 1, 1, 19, 173, 883, 2464, 3514, 2464, 883, 173, 19, 1, 1, 21, 217

Offset: 0

Paul D. Hanna, Jul 24 2003

Determinants of upper left n X n matrices results in A086619: {1,2,10,150,7650,1438200,1051324200,...}, which is the products of the first n terms of the binomial transform of Catalan numbers (A007317): {1,2,5,15,51,188,731,2950,...}.

			Rows begin:
1,_1,__1,__1,___1,____1,____1,_____1, ...
1,_3,__5,__7,___9,___11,___13,____15, ...
1,_5,_14,_28,__47,___71,__100,___134, ...
1,_7,_28,_79,_175,__331,__562,___883, ...
1,_9,_47,175,_504,_1196,_2464,__4572, ...
1,11,_71,331,1196,_3514,_8764,_19244, ...
1,13,100,562,2464,_8764,26172,_67740, ...
1,15,134,883,4572,19244,67740,204831, ...

Cf. A086621 (diagonal), A086622 (antidiagonal sums), A086619 (determinants).

Contribution from Paul Barry, Feb 04 2009: (Start)
T(n,k)=sum{j=0..n+k, C(k,j-k)*C(n+2k-j,k)*if(k<=j,A000108(n-k),0)};
Regarded as a number triangle read by row, columns are generated by sum{j=0..k, C(k,j)*A000108(j)*x^j}*x^k/(1-x)^(k+1). (End)

A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).

Original entry on oeis.org

1, 3, 19, 173, 1881, 22655, 291775, 3940725, 55149025, 793387235, 11668476579, 174735112997, 2656296912361, 40897718776647, 636588467802679, 10002872642155085, 158483629611962025, 2529389028336106475, 40631849127696017275, 656509442594976984405, 10663184061320964941761

Offset: 0

Ilya Gutkovskiy, Nov 29 2021

Cf. A000108, A086618, A086621, A243945.

A349768 := proc(n)
    hypergeom([1/2,-n,n+1],[1,2],-4) ;
    simplify(%) ;
end proc:
seq(A349768(n),n=0..20) ; # R. J. Mathar, Mar 02 2023

Table[Sum[Binomial[n, k] Binomial[n + k, k] Binomial[2 k, k]/(k + 1), {k, 0, n}], {n, 0, 20}]
Table[HypergeometricPFQ[{1/2, -n, n + 1}, {1, 2}, -4], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+k,k)*binomial(2*k,k)/(k+1)); \\ Michel Marcus, Nov 29 2021

From Vaclav Kotesovec, Nov 29 2021: (Start)
D-finite recurrence: n*(n+1)*(2*n - 3)*a(n) = (2*n - 1)*(19*n^2 - 37*n + 12)*a(n-1) - (2*n - 3)*(19*n^2 - 39*n + 14)*a(n-2) + (n-3)*(n-2)*(2*n - 1)*a(n-3).
a(n) ~ sqrt(5) * phi^(6*n + 3) / (8*Pi*n^2), where phi = A001622 is the golden ratio. (End)
D-finite with recurrence n*(n+1)*a(n) +(n+1)*(n-4)*a(n-1) +2*(-171*n^2 +512*n -388)*a(n-2) +2*(9*n^2 +296*n -796)*a(n-3) +(341*n^2 -2425*n +4320)*a(n-4) -19*(n-4)*(n-5)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

cp's OEIS Frontend

A086622 G.f. A(x) satisfies: A(x) = 1/(1-2x) + x^2A(x)^2.

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A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.

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Formula

A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).

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cp's OEIS Frontend

A086622 G.f. A(x) satisfies: A(x) = 1/(1-2*x) + x^2*A(x)^2.

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A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^2.

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Comments

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Formula

A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).

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Maple

Mathematica

PARI

Formula

A086622 G.f. A(x) satisfies: A(x) = 1/(1-2x) + x^2A(x)^2.

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.