A086620 -id:A086620 - cp's OEIS frontend

A086621 Main diagonal of square table A086620 of coefficients, T(n,k), of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.

1, 3, 14, 79, 504, 3514, 26172, 204831, 1664696, 13930840, 119312544, 1041227642, 9228614836, 82867255956, 752405060536, 6897376441167, 63760133568096, 593763928313128, 5565678569009328, 52475976165495960, 497376657383374560, 4736680863568248480, 45304174896889357440

Offset: 0

Paul D. Hanna, Jul 24 2003

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A086620 (table), A086622 (antidiagonal sums).

re:= sumtools:-sumrecursion(binomial(n,j) * binomial(2*n-j,n) * binomial(2*j,j)/(j+1),j,a(n)); # re = Mathar's recurrence
f:= gfun:-rectoproc({re = 0, a(0)=1, a(1)=3, a(2)=14}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Oct 23 2022

a(n) = {sum(j=0, n, binomial(n,j)*binomial(2*n-j, n)*binomial(2*j, j)/(j+1))} \\ Andrew Howroyd, Apr 11 2021

a(n) = Sum_{j=0..n} binomial(n,j) * binomial(2*n-j,n) * binomial(2*j,j)/(j+1). - Andrew Howroyd, Apr 11 2021
D-finite with recurrence n*(n-1)*(n+1)^2*a(n) -2*n*(n-1) *(4*n+3) *(2*n-1) *a(n-1) +4*(n-1) *(16*n^3-20*n^2-13*n+14) *a(n-2) -4*(n-2) *(4*n-9) *(4*n-3) *(n+1) *a(n-3)=0. - R. J. Mathar, Nov 02 2021
a(n) ~ sqrt(5) * 2^(2*n - 3/2) * phi^(2*n + 5/2) / (Pi * n^2), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 19 2021

Terms a(18) and beyond from Andrew Howroyd, Apr 11 2021

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

Original entry on oeis.org

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

A086619 Product of first n terms of the binomial transform of the Catalan numbers (A007317).

Original entry on oeis.org

1, 2, 10, 150, 7650, 1438200, 1051324200, 3101406390000, 37945707181650000, 1966422437567466300000, 438887790263120370963300000, 427664112802721593716655529100000

Offset: 0

Paul D. Hanna, Jul 24 2003

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..56

Cf. A007317, A003046, A086620.

Table[Product[Sum[Binomial[m, k]*Binomial[2*k, k]/(k+1), {k, 0, m}], {m, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 27 2017 *)
Table[Product[Hypergeometric2F1[1/2, -k, 2, -4], {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 27 2017 *)

Determinant of the n X n matrix in the upper left corner of square table A086620.

a(n) ~ c * 5^(n*(n+4)/2) * exp(3*n/2) / (2^(3*n) * Pi^(n/2) * n^(3*n/2 + 27/16)), where c = 0.286457028871544332017553737430712054187435533562040177617845850168... - Vaclav Kotesovec, Oct 27 2017

A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)1, C(n,1)2,..., C(n,k)A051163(k), ..., C(n,n)A051163(n)] and where A051162 equals the antidiagonal sums.

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, 76, 47, 11, 1, 1, 13, 71, 163, 163, 71, 13, 1, 1, 15, 100, 301, 435, 301, 100, 15, 1, 1, 17, 134, 502, 971, 971, 502, 134, 17, 1, 1, 19, 173, 778, 1909, 2577, 1909, 778, 173, 19, 1, 1, 21, 217, 1141

Offset: 0

Paul D. Hanna, Nov 23 2004

Antidiagonal sums form A051163. Main diagonal is A100937. Different from A086620.

			Rows begin:
[1,1,1,1,1,1,1,1,1,...],
[1,3,5,7,9,11,13,15,17,...],
[1,5,14,28,47,71,100,134,...],
[1,7,28,76,163,301,502,778,...],
[1,9,47,163,435,971,1909,3417,...],
[1,11,71,301,971,2577,5917,12167,...],
[1,13,100,502,1909,5917,15678,36744,...],
[1,15,134,778,3417,12167,36744,97272,...],...
Antidiagonal sums form A051163: [1,2,5,12,30,76,194,496,1269,3250,8337,...].
The inverse binomial transform of the rows form the respective rows of the triangle B:
[1*1],
[1*1,1*2],
[1*1,2*2,1*5],
[1*1,3*2,3*5,1*12],
[1*1,4*2,6*5,4*12,1*30],...
where B(n,k) = binomial(n,k)*A051163(k).

Cf. A051163, A100937.

T(n,k)=if(n==0 || k==0,1, sum(j=0,n,binomial(k,j)*binomial(n,j)*sum(i=0,j,T(j-i,i)));)

T(n, k) = Sum_{j=0..n} C(k, j)*C(n, j)*A051162(j), with T(0, 0) = 1 and where Sum_{i=0..n} T(n-i, i) = A051162(n).

cp's OEIS Frontend

A086621 Main diagonal of square table A086620 of coefficients, T(n,k), 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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A086622 G.f. A(x) satisfies: A(x) = 1/(1-2*x) + x^2*A(x)^2.

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A086619 Product of first n terms of the binomial transform of the Catalan numbers (A007317).

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Mathematica

Formula

A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)*1, C(n,1)*2,..., C(n,k)*A051163(k), ..., C(n,n)*A051163(n)] and where A051162 equals the antidiagonal sums.

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A086621 Main diagonal of square table A086620 of coefficients, T(n,k), of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.

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

A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)1, C(n,1)2,..., C(n,k)A051163(k), ..., C(n,n)A051163(n)] and where A051162 equals the antidiagonal sums.