A171422 -id:A171422 - cp's OEIS frontend

A143013 Number of Motzkin n-paths with two kinds of level steps one of which is a final step.

1, 2, 3, 7, 17, 43, 114, 310, 861, 2433, 6970, 20198, 59101, 174373, 518179, 1549545, 4659399, 14079553, 42732230, 130208246, 398174723, 1221573603, 3758835953, 11597578995, 35872937745, 111216324015, 345539568900, 1075693015920

Offset: 0

Michael Somos, Jul 15 2008

nonn

Hankel transform is the (4,-5) Somos-4 variant A171422. - Paul Barry, Dec 08 2009

			A = 1 + (L + F) + (LL + LF + UD) + (LLL + LLF + LUD + UDL + UDF + ULD + UFD) + ...
G.f. = 1 + 2*x + 3*x^2 + 7*x^3 + 17*x^4 + 43*x^5 + 114*x^6 + 310*x^7 + 861*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

Cf. A000108, A007477, A171422.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-x -Sqrt(1-2*x-3*x^2-4*x^3))/(2*x^2) )); // G. C. Greubel, Feb 26 2019

CoefficientList[Series[(1-x -Sqrt[1-2*x-3*x^2-4*x^3])/(2*x^2), {x, 0, 30}], x] (* G. C. Greubel, Feb 26 2019 *)

a(n):=sum(sum((binomial(i-1,k-1)*binomial(k,n-k-i+2)*binomial(k+i-2,i-1))/k,k,1,n-i+2),i,0,n+2); /* Vladimir Kruchinin, May 06 2018 */

{a(n) = if(n<0, 0, polcoeff( (1 - x - sqrt(1 - 2*x - 3*x^2 - 4*x^3 + x^3*O(x^n))) / (2*x^2), n))}

x='x+O('x^30); Vec((1-x-(1-2*x-3*x^2-4*x^3)^(1/2))/(2*x^2)) \\ Altug Alkan, May 06 2018

((1-x -sqrt(1-2*x-3*x^2-4*x^3))/(2*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

Words on alphabet {U,D,L,F} of length n where U is upstep, D is downstep, L and F are level steps and F can only be immediately followed by D or end of word with defining equation A = 1 + F + LA + UADA.
When convolved with itself yields first difference shifted left one place.
G.f. A(x) satisfies: A(x) = 1 + x + A(x)*x + (A(x)*x)^2.
G.f.: (1+x) / (1-x -(x^2 + x^3) / (1-x -(x^2 + x^3) / (1-x -...))).
G.f.: (1 - x - sqrt(1 - 2*x - 3*x^2 - 4*x^3)) / (2*x^2).
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([0,1,2]). - Gary W. Adamson, Oct 27 2008
Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +3*(1-n)*a(n-2) +2*(5-2*n)*a(n-3)=0. - R. J. Mathar, Oct 25 2012
a(n) = Sum_{i=0..n+2} Sum_{k=1..n-i+2} C(i-1,k-1)*C(k,n-k-i+2)*C(k+i-2,i-1)/k. - Vladimir Kruchinin, May 06 2018

A178628 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 - x*y - y = x^3 + x^2 + x.

Original entry on oeis.org

1, 1, -1, -4, -3, 19, 67, -40, -1243, -4299, 25627, 334324, 627929, -29742841, -372632409, 1946165680, 128948361769, 1488182579081, -52394610324649, -2333568937567764, -5642424912729707, 3857844273728205019

Offset: 1

Paul Barry, May 31 2010

a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f.
1/(1-x^2/(1-x^2/(1-4x^2/(1+(3/16)x^2/(1-(76/9)x^2/(1-(201/361)x^2/(1-... where
1,4,-3/16,76/9,201/361,... are the x-coordinates of the multiples of z=(0,0)
on E:y^2-xy-y=x^3+x^2+x.

G. C. Greubel, Table of n, a(n) for n = 1..160
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.

Cf. A006720.

(p, q) Somos-4 sequences: A171422, A174168, A174170, A174404, A174809, A174811, A174882, A178075, A178077, A178081, A178079, A178376, A178377, A178384, A178417, A178418, A178621, A178622, A178624, A178625, A178627, A178628, A178644, A184019, A184121, A188313, A188315, A352625.

I:=[1,1,-1,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 18 2018

RecurrenceTable[{a[n] == (a[n-1]*a[n-3] +a[n-2]^2)/a[n-4], a[1] == 1, a[2] == 1, a[3] == -1, a[4] == -4}, a, {n,1,30}] (* G. C. Greubel, Sep 18 2018 *)

a(n)=local(E,z);E=ellinit([ -1,1,-1,1,0]);z=ellpointtoz(E,[0,0]); round(ellsigma(E,n*z)/ellsigma(E,z)^(n^2))

m=30; v=concat([1,1,-1,-4], vector(m-4)); for(n=5, m, v[n] = ( v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 18 2018

{a(n) = subst(elldivpol(ellinit([-1, 1, -1, 1, 0]), n), x ,0)}; /* Michael Somos, Jul 05 2024 */

@CachedFunction
def a(n): # a = A178628
    if n<5: return (0,1,1,-1,-4)[n]
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 05 2024

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>4.

a(n) = -a(-n). a(n) = (-a(n-1)*a(n-4) +4*a(n-2)*a(n-3))/a(n-5) for all n in Z except n=5. - Michael Somos, Jul 05 2024

Offset changed to 0. - Michael Somos, Jul 05 2024

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A143013 Number of Motzkin n-paths with two kinds of level steps one of which is a final step.

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A178628 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 - x*y - y = x^3 + x^2 + x.

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