A167422 -id:A167422 - cp's OEIS frontend

A182959 Expansion of o.g.f. 2(1+x)^2/(1-2x+sqrt(1-8*x)).

1, 5, 20, 96, 528, 3136, 19584, 126720, 841984, 5710848, 39376896, 275185664, 1944821760, 13875707904, 99807723520, 722997411840, 5269761884160, 38620004352000, 284405842575360, 2103530005463040, 15619068033761280

Offset: 0

Paul D. Hanna, Dec 31 2010

			G.f.: A(x) = 1 + 5*x + 20*x^2 + 96*x^3 + 528*x^4 + 3136*x^5 +...
where A(x*F(x)^3) = F(x) is the g.f. of A182960:
F(x) = 1 + 5*x + 95*x^2 + 2496*x^3 + 76063*x^4 + 2524161*x^5 +...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A182960, A001450, A167422, A372215.

CoefficientList[ Series[2 (1 + x)^2/(1 - 2 x + Sqrt[1 - 8 x]), {x, 0, 20}], x]  (* Robert G. Wilson v, Dec 31 2010 *)

{a(n)=polcoeff(2*(1+x)^2/(1-2*x+sqrt(1-8*x+x*O(x^n))),n)}

Let F(x) be the g.f. of A182960, then g.f. of this sequence satisfies:
* A(x) = F(x/A(x)^3) and A(x*F(x)^3) = F(x);
* A(x) = [x/Series_Reversion( x*F(x)^3 )]^(1/3).
G.f.: 1/2/x - 1/2 - x - (1+x)/x/G(0), where G(k)= 1 + 1/(1 - 4*x*(2*k+1)/(4*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
a(n) ~ 9*2^(3*n-2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013
From Peter Bala, Oct 04 2015: (Start)
O.g.f. A(x) = (1 + x)*(2*C(2*x) - 1), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.
[x^n] A(x)^(3*n) = binomial(6*n,2*n). Cf. with the identity [x^n] ( (1 + x)*C(x) )^(5*n) = binomial(5*n,2*n) = A001450(n). (End)
Conjecture: D-finite with recurrence (n+1)*a(n) +(-7*n+3)*a(n-1) +4*(-2*n+5)*a(n-2)=0. - R. J. Mathar, Jan 22 2020
From Peter Bala, May 15 2023: (Start)
a(n) = 3*(2^n)*(3*n - 1)/(n*(n + 1)) * binomial(2*n-2,n-1) for n >= 2.
(n + 1)*(3*n - 4)*a(n) = 4*(2*n - 3)*(3*n - 1)*a(n-1) for n >= 3 with a(2) = 20. Mathar's conjectured second order recurrence above follows from this. (End)
[x^n] A(x)^n = A372215(n). - Peter Bala, Nov 07 2024

A381787 G.f. A(x) satisfies A(x) = (1 + x) * C(x*A(x)), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 5, 22, 112, 623, 3664, 22405, 141002, 907228, 5940663, 39459873, 265228359, 1800608563, 12328843910, 85040632504, 590371016188, 4121775003434, 28921911896836, 203854515625194, 1442669458817907, 10247020573880520, 73024240955785936, 521973882076798493

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A167422, A381785, A381786.

Cf. A000108.

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

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

A167423 Hankel transform of a simple Catalan convolution.

Original entry on oeis.org

1, -1, -11, -50, -186, -631, -2029, -6299, -19075, -56704, -166164, -481391, -1381691, -3935125, -11134331, -31328366, -87721614, -244588519, -679429225, -1881102959, -5192705779, -14296088956, -39263958696, -107601905375, -294291714551, -803416991401

Offset: 0

Paul Barry, Nov 03 2009

Hankel transform of A167422.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A167422, A000045, A000032.

[Fibonacci(2*n)*(1-3*n)/2 + Lucas(2*n)*(1-n)/2: n in [0..30]]; // Vincenzo Librandi, Jun 13 2016

Table[((1-3n) Fibonacci[2n] + (1-n) LucasL[2n])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)
LinearRecurrence[{6, -11, 6, -1}, {1, -1, -11, -50}, 50] (* G. C. Greubel, Jun 12 2016 *)

Vec((1-7*x+6*x^2-x^3)/(1-6*x+11*x^2-6*x^3+x^4) + O(x^100)) \\ Altug Alkan, Oct 29 2015

G.f.: ( 1-7*x+6*x^2-x^3 ) / (x^2-3*x+1)^2 .
a(n) = F(2*n)*(1-3*n)/2 + L(2*n)*(1-n)/2. - Paul Barry, Feb 22 2010
a(n) = 3*A001871(n-1) - 2*A001871(n) + F(2*n+4). - Ralf Stephan, May 21 2014
a(n) = 1 - Sum_{k=1..n} k*F(2*k+1), where F(n) = A000045(n). - Vladimir Reshetnikov, Oct 28 2015

A259550 a(n) = C(5n-1,2n)/3, n > 0, a(0) = 1.

Original entry on oeis.org

1, 2, 42, 1001, 25194, 653752, 17298645, 463991880, 12570420330, 343176898988, 9425842448792, 260170725132045, 7210477496434485, 200519284375732896, 5592628786362932776, 156375886125188595376, 4382048530314336892010, 123033460966787345446836

Offset: 0

Vladimir Kruchinin, Jun 30 2015

D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
V. V. Kruchinin and D. V. Kruchinin, Composita and its properties, J. Analysis and Number Theory 2 (2014), 1-8.
V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
D. V. Kruchinin, On solving some functional equations, Advances in Difference Equations, Vol. 1 (2015), 1687-1847.

Cf. A000108, A167422, A001450, A060941.

[1] cat [Binomial(5*n-1, 2*n)/3: n in [1..20]]; // Vincenzo Librandi, Jul 01 2015

Join[{1}, Table[Binomial[5 n - 1, 2 n]/3, {n, 30}]] (* Vincenzo Librandi, Jul 01 2015 *)

makelist(if n=0 then 1 else binomial(5*n-1,2*n)/3,n,0,20);

vector(20, n, n--; if (n==0, 1, binomial(5*n-1,2*n)/3)) \\ Michel Marcus, Jul 01 2015

G.f.: A(x) = 1 + (x*B(x)')/(B(x)), B(x) = (1 + x*B(x)^5)*C(x*B(x)^5) is g.f. of A060941, C(x) is g.f. of Catalan numbers.
a(n) = n*Sum_{i = 0..n}((C(5*n,i)*C(7*n-2*i-1,n-i))/(6*n-i)), n > 1, a(0) = 1.
a(n) = 1/5*A001450(n) for n >= 1. exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 23*x^2 + 377*x^3 + ... is the o.g.f. for the sequence of Duchon numbers A060941. - Peter Bala, Oct 05 2015
D-finite with recurrence 6*n*(3*n-1)*(2*n-1)*(3*n-2)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

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A182959 Expansion of o.g.f. 2*(1+x)^2/(1-2*x+sqrt(1-8*x)).

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A381787 G.f. A(x) satisfies A(x) = (1 + x) * C(x*A(x)), where C(x) is the g.f. of A000108.

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A167423 Hankel transform of a simple Catalan convolution.

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A259550 a(n) = C(5*n-1,2*n)/3, n > 0, a(0) = 1.

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A182959 Expansion of o.g.f. 2(1+x)^2/(1-2x+sqrt(1-8*x)).

A259550 a(n) = C(5n-1,2n)/3, n > 0, a(0) = 1.