A143013 -id:A143013 - cp's OEIS frontend

A171422 A (4,-5) Somos-4 sequence.

1, -1, -9, -41, -241, -271, 27329, 651521, 18425809, 399122991, -24055184809, -3943757411849, -498726356978849, -74609317288333151, 2771780972000481601, 8459493968101503667649, 5139175203350482789079649

Offset: 0

Paul Barry, Dec 08 2009

Hankel transform of A143013.

G. C. Greubel, Table of n, a(n) for n = 0..109

I:=[1,-1,-9,-41]; [n le 4 select I[n] else (4*Self(n-1)*Self(n-3) -5*Self(n-2)^2)/Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 18 2018

RecurrenceTable[{a[n] == (4*a[n-1]*a[n-3] -5*a[n-2]^2)/a[n-4], a[0] == 1, a[1] == -1, a[2] == -9, a[3] == -41}, a, {n, 0, 30}] (* G. C. Greubel, Sep 18 2018 *)

m=30; v=concat([1,-1,-9,-41], vector(m-4)); for(n=5, m, v[n] = ( 4*v[n-1]*v[n-3] -5*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 18 2018

a(n) = (4*a(n-1)*a(n-3) - 5*a(n-2)^2)/a(n-4), n>=3.

A253918 Number of Motzkin n-paths with two kinds of level steps both of which are final steps.

Original entry on oeis.org

1, 2, 1, 4, 6, 12, 29, 56, 134, 300, 682, 1624, 3772, 9016, 21597, 51888, 126086, 306636, 750398, 1843864, 4543604, 11242088, 27887730, 69378192, 173050396, 432596216, 1083862132, 2720961520, 6843557944, 17242789104, 43514507997, 109983815264, 278385212358

Offset: 0

Michael Somos, Jan 19 2015

nonn

Number of words of length n on alphabet {U,D,L,R} where U is upstep, D is downstep, L and R are level steps and can only be immediately followed by D or end of word. Defining equation is A = 1 + L + R + UADA.

			G.f. = 1 + 2*x + x^2 + 4*x^3 + 6*x^4 + 12*x^5 + 29*x^6 + 56*x^7 + ...
A = 1 + (L + R) + (UD) + (ULD + URD + UDR + UDL) + (UDUD + UUDD + ULDL + ULDR + URDL + URDR) + ...

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

Cf. A143013.

a[ n_] := SeriesCoefficient[ (1 - Sqrt[1 - 4 x^2 - 8x^3]) / (2 x^2), {x, 0, n}];

a(n):=sum((binomial(k+1,n-2*k)*binomial(2*k,k)*2^(n-2*k))/(k+1),k,0,n); /* Vladimir Kruchinin, Mar 11 2016 */

{a(n) = if( n<0, 0, polcoeff( (1 - sqrt(1 - 4*x^2 - 8*x^3  + x^3*O(x^n))) / (2*x^2), n))};

{a(n) = my(A); if( n<0, 0, A = O(x); for(k=0, n\2, A = 1 + 2*x + sqr(x*A)); polcoeff( A, n))};

When convolved with itself yields sequence shifted left two places.
G.f. A(x) satisfies A(x) = 1 + 2*x + (A(x)*x)^2.
G.f.: (1 - sqrt(1 - 4*x^2 - 8*x^3)) / (2*x^2).
0 = a(n)*(8*n+4) + a(n+1)*(4*n+8) + a(n+3)*(-n-5) for n>=-1.
a(n) = Sum_{k=0..n}((binomial(k+1,n-2*k)*binomial(2*k,k)*2^(n-2*k))/(k+1)). - Vladimir Kruchinin, Mar 11 2016
a(n) ~ sqrt(3-4*c^2) * 4^n * c^n * (1+2*c)^(n+1) / (sqrt(Pi)*n^(3/2)), where c = 0.37743883312334638... is the root of the equation 4*c^2*(1 + 2*c) = 1. - Vaclav Kotesovec, Mar 10 2016

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A171422 A (4,-5) Somos-4 sequence.

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A253918 Number of Motzkin n-paths with two kinds of level steps both of which are final steps.

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