A200376 -id:A200376 - cp's OEIS frontend

A200375 Product of Catalan and Jacobsthal numbers: a(n) = A000108(n)*A001045(n+1).

1, 1, 6, 25, 154, 882, 5676, 36465, 244530, 1657942, 11471668, 80242890, 568080772, 4056976900, 29212908120, 211783889025, 1544811959970, 11328491394990, 83473572128100, 617702666484750, 4588654943721420, 34206312386929020, 255803818897858920, 1918528298674328250, 14427334095935095764

Offset: 0

Paul D. Hanna, Nov 16 2011

nonn

More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |b|>0, |c|>0, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.

			G.f.: A(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 + 429*85*x^7 + 1430*171*x^8 +...+ A000108(n)*A001045(n)*x^n +...
The g.f. of the Jacobsthal sequence A001045, F(x) = 1/(1-x-2*x^2), begins:
F(x) = 1 + x + 3*x^2 + 5*x^3 + 11*x^4 + 21*x^5 + 43*x^6 + 85*x^7 + 171*x^8 +...
The g.f. of A200376, where G(x) =  A(x/G(x)), begins:
G(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...
in which the odd-indexed coefficients are powers of 9.

Michael De Vlieger, Table of n, a(n) for n = 0..1112
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Paul Barry and Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Cf. A200376, A098614, A098616, A200312.

Array[CatalanNumber[# - 1] (2^# - (-1)^#)/3 &, 25] (* Michael De Vlieger, Apr 24 2018 *)

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

{a(n) = polcoef(sqrt((1-2*x - sqrt(1-4*x-32*x^2 +O(x^(n+3))))/2)/(3*x), n)}

{a(n) = polcoef((1/x)*serreverse(x-x^2 - 4*x^3*sum(m=0,n\2,binomial(2*m,m)/(m+1)*3^m*x^(2*m)) +x^3*O(x^n)), n)}

G.f.: sqrt( (1-2*x - sqrt(1-4*x-32*x^2))/2 )/(3*x).
G.f.: (1/x)*Series_Reversion(x-x^2 - 4*x^3*Sum_{n>=0} A000108(n)*3^n*x^(2*n) ).
G.f. satisfies: A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) is the g.f. of A200376: G(x) = 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2).
n*(n+1)*a(n) -2*n*(2*n-1)*a(n-1) -8*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 17 2011
a(n) = binomial(2*n,n)/(n+1) * (2^(n+1) + (-1)^n)/3.
From Peter Bala, Aug 17 2021: (Start)
G.f.: A(x) = (sqrt(1 + 4*x) - sqrt(1 - 8*x))/(6*x).
A(x) = 1/sqrt(1 + 4*x)*c( 3*x/(1 + 4*x) ), where c(x) = (1 - sqrt(1- 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Cf. A151374.
In general, [x^n] ( 1/sqrt(1 + 4*x)*c( k*x/(1 + 4*x) ) ) = Catalan(n)*((k-1)^(n+1) + (-1)^(n+1))/k.
A(x) = 1/sqrt(1 - 8*x)*c( -3*x/(1 - 8*x) ). (End)
G.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*A(x)^2 + 9*x^2*A(x)^4 ). - Paul D. Hanna, Dec 14 2024

Typo in Name corrected by Peter Bala, Aug 17 2021

A277604 Array of coefficients T(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = sqrt(1 + 2xA(k,x) + (4k+1)x^2*(A(k,x))^2), k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 1, 7, 9, 13, 1, 1, 1, 9, 13, 37, 25, 1, 1, 1, 11, 17, 73, 81, 61, 1, 1, 1, 13, 21, 121, 169, 301, 125, 1, 1, 1, 15, 25, 181, 289, 841, 729, 295, 1, 1, 1, 17, 29, 253, 441, 1801, 2197, 2549, 625, 1, 1, 1, 19, 33, 337, 625, 3301, 4913, 10123, 6561, 1447, 1

Offset: 0

Werner Schulte, Oct 29 2016

For k = 0 see A000012, for k = 1 see A098615, and for k = 2 see A200376.
It will be interesting using the formulae for k < 0 (attention: signed terms!). Especially for k = -1 see A157674.
If G is the g.f. of central binomial coefficients (see A000984) and B(k,x) = G(k*x^2), then B(k,x) = A(k,x)/(1+x*A(k,x)) and A(k,x) = B(k,x) / (1-x*B(k,x)) for k >= 0. - Werner Schulte, Aug 07 2017

			The terms define the array T(k,n) for k >= 0 and n >= 0, i.e.,
k\n  0  1   2   3    4     5      6      7       8        9  . . .
0:   1  1   1   1    1     1      1      1       1        1  . . .
1:   1  1   3   5   13    25     61    125     295      625  . . .
2:   1  1   5   9   37    81    301    729    2549     6561  . . .
3:   1  1   7  13   73   169    841   2197   10123    28561  . . .
4:   1  1   9  17  121   289   1801   4913   28057    83521  . . .
5:   1  1  11  21  181   441   3301   9261   63071   194481  . . .
6:   1  1  13  25  253   625   5461  15625  123565   390625  . . .
7:   1  1  15  29  337   841   8401  24389  219619   707281  . . .
8:   1  1  17  33  433  1089  12241  35937  362993  1185921  . . .
9:   1  1  19  37  541  1369  17101  50653  567127  1874161  . . .
etc.

Cf. A000012, A000045, A000108, A000984, A015441, A046521, A098615, A111959, A200376.

A(k,x) = (x + sqrt(1 - 4*k*x^2))/(1 - (4*k+1)*x^2) for k >= 0.
T(k,0) = 1 and T(k,2*n+2) = (4*k+1)^(n+1)-2*(Sum_{i=0..n} A000108(i)*k^(i+1)* (4*k+1)^(n-i)), and T(k,2*n+1) = (4*k+1)^n for k >= 0 and n >= 0.
A(k,x) = 1/(1 - x - 2*k*x^2*C(k*x^2)), k >= 0, where C is the g.f. of A000108.
Conjecture: If B(k,n) satisfy B(k,0) = B(k,1) = 1 and B(k,n+2) = B(k,n+1) + k*B(k,n) for k >= 0 and n >= 0 (generalized Fibonacci numbers, see A015441) and G(k,x) = Sum_{n>=0} A000108(n)*B(k,n)*x^n for k >= 0, then you will have (1): A(k,x*G(k,x)) = G(k,x) and (2): G(k,x/A(k,x)) = A(k,x) for k >= 0. Especially for k = 1 see A098615 and for k = 2 see A200376.
Conjecture: T(k,2*n) = Sum_{i=0..n} A046521(n,i)*k^(n-i) for k, n >= 0. - Werner Schulte, Aug 02 2017
Recurrence: T(k,2*n+2) = (4*k+1)*T(k,2*n)-2*k^(n+1)*A000108(n) with initial value T(k,0) = 1 for k >= 0 and n >= 0. - Werner Schulte, Aug 09 2017
T(k,n) = Sum_{i=0..n} A111959(n,i)*k^((n-i)/2) for k >= 0 and n >= 0. - Werner Schulte, Aug 09 2017

cp's OEIS Frontend

A200375 Product of Catalan and Jacobsthal numbers: a(n) = A000108(n)*A001045(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A277604 Array of coefficients T(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = sqrt(1 + 2xA(k,x) + (4k+1)x^2*(A(k,x))^2), k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

cp's OEIS Frontend

A200375 Product of Catalan and Jacobsthal numbers: a(n) = A000108(n)*A001045(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A277604 Array of coefficients T(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = sqrt(1 + 2*x*A(k,x) + (4*k+1)*x^2*(A(k,x))^2), k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A277604 Array of coefficients T(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = sqrt(1 + 2xA(k,x) + (4k+1)x^2*(A(k,x))^2), k >= 0.