A056010 -id:A056010 - cp's OEIS frontend

A025262 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 4.

1, 1, 1, 3, 8, 23, 68, 207, 644, 2040, 6558, 21343, 70186, 232864, 778550, 2620459, 8872074, 30195288, 103246502, 354508628, 1221846856, 4225644866, 14659644348, 51002664023, 177909901566, 622093882290, 2180123564130, 7656055966092

Offset: 1

Clark Kimberling

nonn

a(n) is the number of ascent sequences (A022493) of length n-1 such that the nonzero entries are weakly increasing and no two consecutive entries are both 0. For example a(4) = 3 counts 010, 011, 012 and a(5) = 8 counts 0101, 0102, 0110, 0111, 0112, 0120, 0122, 0123. - David Callan, Nov 25 2021

The o.g.f. y (= x + x^2 + x^3 + ...) of this sequence satisfies y^2 - y = x^3 - x. If y is replaced by -y, then it is the elliptic curve y^2 + y = x^3 - x with LMFDB label 37.a1 (Cremona label 37a1) associated to the Somos-4 sequence via elliptic divisibility sequence A006769. - Michael Somos, Apr 18 2023

			G.f. = x + x^2 + x^3 + 3*x^4 + 8*x^5 + 23*x^6 + 68*x^7 + 207*x^8 + 644*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1766
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Michael Somos, Number Walls in Combinatorics.
Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], 2014; see p. 7.

Cf. A000108, A176703, A056010, A025268, A006769.

nmax = 30; aa = ConstantArray[0, nmax]; aa[[1]] = 1; aa[[2]] = 1; aa[[3]] = 1; Do[aa[[n]] = Sum[aa[[k]] * aa[[n - k]], {k, 1, n - 1}], {n, 4, nmax}]; aa (* Vaclav Kotesovec, Jan 25 2015 *)
Nest[Append[#, #.Reverse[#]] &, {1, 1, 1}, 25] (* Jan Mangaldan, Jul 07 2020 *)

{a(n) = polcoeff( (1 - sqrt(1 - 4*x + 4*x^3 + x * O(x^n))) / 2, n)}; /* Michael Somos, Aug 04 2000 */

G.f.: (1 - sqrt(1 - 4*x + 4*x^3)) / 2. Satisfies A(x) - A(x)^2 = x - x^3. - Michael Somos, Aug 04 2000
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example Phi([1]) is the Catalan numbers A000108. The present sequence is Phi([1,1,1]). - Gary W. Adamson, Oct 27 2008
Row sums of A176703 if offset 0. - Michael Somos, Jan 09 2012
a(n+2) = A056010(n) if n >= 0.
a(n) = Sum_{m=0..floor((n-1)/2)} C(n-2*m-1)*binomial(n-2*m,m)*(-1)^m, where C = A000108 are the Catalan numbers. - Vladimir Kruchinin, Jan 26 2013
0 = a(n)*(+16*a(n+1) - 64*a(n+3) + 22*a(n+4)) + a(n+1)*(+32*a(n+2) - 14*a(n+3)) + a(n+2)*(+16*a(n+3) - 10*a(n+4)) + a(n+3)*(+2*a(n+3) + a(n+4)) if n>0. - Michael Somos, Jan 18 2015
Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 2*(2*n-9)*a(n-3). - Vaclav Kotesovec, Jan 25 2015
a(n) ~ sqrt(3 - 8*r) * (4 - 4*r^2)^n / (4*sqrt(Pi)*n^(3/2)), where r = 2*sin(arccos(-3^(3/2)/8)/3 - Pi/6)/sqrt(3). - Vaclav Kotesovec, Jun 05 2022

A160702 Sequence such that the Hankel transform of a(n+1) satisfies a generalized Somos-4 recurrence.

Original entry on oeis.org

1, 1, 1, 5, 19, 79, 333, 1441, 6351, 28451, 129185, 593373, 2752427, 12876343, 60684533, 287857209, 1373286375, 6584979659, 31719337353, 153416338549, 744777567043, 3627787084319

Offset: 0

Paul Barry, May 24 2009

The Hankel transform of a(n+1) satisfies a generalized Somos-4 Hankel determinant recurrence.
Hankel transform of a(n+1) is A160703. In general, we can conjecture that the Hankel transform of
h(n) of a(n+1), where a(n)=if(n=0,1,if(n=1,1,if(n=2,1,r*a(n-1)+s*sum{k=0..n-2, a(k)*a(n-1-k)}))),
satisfies the generalized Somos-4 recurrence
h(n)=(s^2*h(n-1)*h(n-3)+s^3*(2*s+r-2)*h(n-2)^2)/h(n-4).
The case r=s=1 is proved in the Xin reference.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Gouce Xin, Proof of the Somos-4 Hankel determinants conjecture, Advances in Applied Mathematics, Volume 42, Issue 2, February 2009, Pages 152-156.

Cf.: A025262, A056010, A006720.

CoefficientList[Series[1/4+(1-Sqrt[16*x^3+x^2-6*x+1])/(4*x),{x,0,20}],x] (* Vaclav Kotesovec, Nov 20 2012 *)

a(n) = if(n=0,1,if(n=1,1,if(n=2,1,a(n-1)+2*sum{k=0..n-2, a(k)*a(n-1-k)})))
Recurrence: (n+1)*a(n) = 3*(2*n-1)*a(n-1) - (n-2)*a(n-2) - 8*(2*n-7)*a(n-3). - Vaclav Kotesovec, Nov 20 2012
G.f.: 1/4+(1-sqrt(16*x^3+x^2-6*x+1))/(4*x). - Vaclav Kotesovec, Nov 20 2012

A173992 Sequence whose Hankel transform is the Somos (4) sequence.

Original entry on oeis.org

1, 1, 3, 6, 15, 34, 83, 198, 488, 1202, 3015, 7608, 19432, 49994, 129779, 339176, 892600, 2362634, 6288156, 16816232, 45170466, 121812152, 329679487, 895171236, 2437885058, 6657311202, 18224979884, 50006899724, 137502724754

Offset: 0

Paul Barry, Mar 04 2010

Hankel transform is A006720(n+3). In general, the sequence with g.f. ((1-x)/(1-(r+1)*x))*c(x^2*(1-x)/(1-(r+1)*x)) will have a Somos (1,r) Hankel transform.

a(n) is the number of rooted plane 2-trees with integer compositions labeling the leaves (empty labels are allowed), with total size n. The total size is the number of edges in the tree plus the sum of the sizes of the integer compositions labeling the leaves. Example; a(2)=3 because there are two trees that consist of the root and no descendants, hence the root is itself a leaf and it can be labeled by either 2=2 or by 2=1+1, and then there is the tree that consists of the root with two descendants and no labels on the two leaves. - Ricardo Gómez Aíza, Feb 26 2024

G. C. Greubel, Table of n, a(n) for n = 0..2000
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024.

Cf. A000108, A006720, A056010.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-2*x-Sqrt((1-2*x)*(1-2*x-4*x^2+4*x^3)))/(2*x^2*(1-2*x)))); // G. C. Greubel, Sep 25 2018

with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:=(1-2*x-sqrt((1-2*x)*(1-2*x-4*x^2+4*x^3)))/(2*x^2*(1-2*x)):
req1:= FindRE(ogf,x,u(n)); inits:= {seq(u(i-1)=[1, 1, 3, 6, 15, 34][i],i=1..6)}:
req2:= subs(n=n-4, MinimalRecurrence(req1,u(n),inits)[1]); # Mathar's recurrence
a:= gfun:-rectoproc({req2} union inits, u(n), remember):
seq(a(n),n=0..28); # Georg Fischer, Nov 03 2022

A173992[n_] := Sum[CatalanNumber[k] Sum[Binomial[k + 1, i] Binomial[n - k - i, n - 2 k - i] (-1)^i Floor[2^(n - 2 k - i)], {i, 0, k + 1}], {k, 0, Floor[n/2]}] (* Eric Rowland, May 15 2017 *)
CoefficientList[Series[(1-2*x -Sqrt[(1-2*x)*(1-2*x-4*x^2+4*x^3)])/(2*x^2* (1-2*x)), {x, 0, 50}], x] (* G. C. Greubel, Sep 25 2018 *)

a(n) = sum(k=0, n\2, binomial(2*k,k)/(k+1)*sum(i=0, k+1, binomial(k+1,i)*binomial(n-k-i,n-2*k-i)*(-1)^i*2^(n-2*k-i))); \\ Michel Marcus, May 15 2017

x='x+O('x^50); Vec((1-2*x-((1-2*x)*(1-2*x-4*x^2+4*x^3))^(1/2))/(2*x^2*(1-2*x))) \\ Altug Alkan, Sep 25 2018

G.f.: ((1-x)/(1-2*x)) * c(x^2*(1-x)/(1-2*x)) = (1-2*x-sqrt((1-2x)*(1-2*x-4*x^2+4*x^3)))/(2*x^2*(1-2*x)), c(x) the g.f. of A000108;
a(n) = Sum_{k=0..floor(n/2), A000108(k)*Sum_{i=0..k+1, C(k+1,i)*C(n-k-i,n-2k-i)*(-1)^i*2^(n-2k-i)}}.
D-finite with recurrence: (n+2)*a(n) -4*(n+1)*a(n-1) +4*a(n-2) +2*(6n-11)*a(n-3) +8*(3-n)*a(n-4)=0. - R. J. Mathar, Nov 17 2011
a(n) ~ sqrt(2-5*c+4*c^2)/(2*c*(1-2*c)*sqrt(Pi*n^3))*(1/c)^n where c=(4+(1+i*sqrt(3))*(1+3*i*sqrt(111))^(1/3)+80/((sqrt(3)+i)^2*(1+3*i*sqrt(111))^(1/3)))/12. - Ricardo Gómez Aíza, Feb 26 2024

A157002 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.

Original entry on oeis.org

1, 0, 1, 2, 6, 17, 51, 156, 488, 1552, 5006, 16337, 53849, 179015, 599535, 2020924, 6851150, 23344138, 79902364, 274606264, 947240592, 3278404274, 11381240074, 39621423949, 138288477617, 483805404673, 1696318159457, 5959737806635

Offset: 0

Paul Barry, Feb 20 2009

Image of the Catalan numbers A000108 by the Riordan array (1-x,x(1-x^2)). Hankel transform is A006720(n+1).
The sequence a(n)+a(n+1) begins 1,1,3,8,23,68,... which is A056010. The sequence a(n)+a(n-1) begins 1,1,1,3,8,23,68,... which is A025262. This is obtained by applying (1-x^2,x(1-x^2)) to the Catalan numbers.
Hankel transform of a(n+1) is -A051138(n). - Michael Somos, Feb 10 2015

			G.f. = 1 + x^2 + 2*x^3 + 6*x^4 + 17*x^5 + 51*x^6 + 156*x^7 + 488*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Gouce Xin, Proof of the Somos-4 Hankel determinants conjecture, Advances in Applied Mathematics, Volume 42, Issue 2, February 2009, Pages 152-156.

Cf. A000108, A006720, A025262, A051138, A056010.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1 -Sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)) )); // G. C. Greubel, Feb 26 2019

CoefficientList[Series[(1-Sqrt[1-4x(1-x^2)])/(2x(1+x)), {x,0,30}], x] (* G. C. Greubel, Feb 26 2019 *)

{a(n) = if( n<0, -(-1)^n / 2 * (n<-1), polcoeff( (1 - sqrt(1 - 4*x * (1 - x^2) + x^2 * O(x^n))) / (2 * x * (1 + x)), n))}; /* Michael Somos, Feb 10 2015 */

((1-sqrt(1-4*x*(1-x^2)))/(2*x*(1+x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

G.f.: (1 - sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)).
a(n) = Sum_{k=0..n} (-1)^floor((n-k+1)/2)*C(k,floor((n-k)/2))*A000108(k).
Conjecture: (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-2*n+1)*a(n-2) +2*(2*n-7)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 19 2014
0 = a(n)*(+16*a(n+1) + 16*a(n+2) - 64*a(n+3) - 42*a(n+4) + 22*a(n+5)) + a(n+1)*(+16*a(n+1) + 48*a(n+2) - 46*a(n+3) - 56*a(n+4) + 22*a(n+5)) + a(n+2)*(+32*a(n+2) + 34*a(n+3) - 8*a(n+4) - 10*a(n+5)) + a(n+3)*(+18*a(n+3) + 11*a(n+4) - 9*a(n+5)) + a(n+4)*(+3*a(n+4) + a(n+5)) for all n in Z. - Michael Somos, Feb 10 2015

A087626 Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).

Original entry on oeis.org

1, 2, 5, 13, 36, 104, 311, 955, 2995, 9553, 30896, 101082, 333946, 1112496, 3732955, 12605029, 42800317, 146046819, 500555447, 1722402303, 5948047169, 20607691517, 71610355540, 249520257106, 871614139396, 3051737703526

Offset: 0

Michael Somos, Sep 16 2003

nonn

			G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 36*x^4 + 104*x^5 + 311*x^6 + 955*x^7 + ... - _Michael Somos_, Mar 28 2020

Robert Israel, Table of n, a(n) for n = 0..1762
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.

Cf. A006720, A051138, A056010, A157003.

f:= gfun:-rectoproc({(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 13},a(n),remember):
map(f, [$0..50]); # Robert Israel, Oct 26 2018

CoefficientList[Series[2/(1-2x+Sqrt[1-4x+4x^3]),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2017 *)

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

G.f.: 2/(1-2x+sqrt(1-4x+4x^3)).
G.f. A(x) satisfies 0 = x^2*(1-x)*A(x)^2 - (1-2*x)*A(x) + 1.
First backwards difference is A056010.
(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4)=0. - Robert Israel, Oct 26 2018
HANKEL transform is A006720(n+2). HANKEL transform with 0 prepended is -A051138.
INVERT transform of A157003. - Michael Somos, Mar 28 2020

A349185 G.f. A(x) satisfies: A(x) = (1 - x) / (1 - 2 * x - x^2 - x^2 * A(x)).

Original entry on oeis.org

1, 1, 4, 11, 35, 111, 365, 1221, 4160, 14371, 50251, 177503, 632514, 2271027, 8208259, 29840993, 109049568, 400352639, 1475929092, 5461571729, 20279092033, 75531360153, 282123848574, 1056539226257, 3966214054639, 14922195004703, 56258116929483, 212505815364639, 804142811583006

Offset: 0

Ilya Gutkovskiy, Nov 09 2021

nonn

Cf. A056010, A085139, A086581, A128720, A349186.

nmax = 28; A[] = 0; Do[A[x] = (1 - x)/(1 - 2 x - x^2 - x^2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 28; CoefficientList[Series[(1 - 2 x - x^2 - Sqrt[1 - 4 x - 2 x^2 + 8 x^3 + x^4])/(2 x^2), {x, 0, nmax}], x]
a[0] = a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]

G.f.: (1 - 2*x - x^2 - sqrt(1 - 4*x - 2*x^2 + 8*x^3 + x^4)) / (2*x^2).

a(0) = a(1) = 1; a(n) = 2 * a(n-1) + a(n-2) + Sum_{k=0..n-2} a(k) * a(n-k-2).

cp's OEIS Frontend

A025262 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.

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A160702 Sequence such that the Hankel transform of a(n+1) satisfies a generalized Somos-4 recurrence.

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A173992 Sequence whose Hankel transform is the Somos (4) sequence.

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A157002 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.

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A087626 Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).

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A349185 G.f. A(x) satisfies: A(x) = (1 - x) / (1 - 2 * x - x^2 - x^2 * A(x)).

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A025262 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 4.