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A191354 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), and (2,1). 4
1, 1, 3, 9, 25, 75, 227, 693, 2139, 6645, 20757, 65139, 205189, 648427, 2054775, 6526841, 20775357, 66251247, 211617131, 676930325, 2168252571, 6953348149, 22322825865, 71735559255, 230735316795, 742773456825, 2392949225565, 7714727440755, 24888317247705, 80341227688095 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.

FORMULA

G.f.: 1/sqrt(1-2*x-3*x^2-4*x^3). - Mark van Hoeij, Apr 16 2013

G.f.: Q(0), where Q(k) = 1 + x*(2+3*x+4*x^2)*(4*k+1)/( 4*k+2 - x*(2+3*x+4*x^2)*(4*k+2)*(4*k+3)/(x*(2+3*x+4*x^2)*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013

a(n) = Sum_{k=0..n} (binomial(2*k,k) * Sum_{j=0..k} (binomial(j,n-k-j) *binomial(k,j)*2^(j-k)*3^(-n+k+2*j)*4^(n-k-2*j))). - Vladimir Kruchinin, Feb 27 2016

D-finite: +(n)*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jan 14 2020

MATHEMATICA

a[n_]:= Sum[Binomial[2k, k]*Sum[Binomial[j, n-k-j]*Binomial[k, j]*2^(j-k) *3^(-n+k+2j)*4^(n-k-2j), {j, 0, k}], {k, 0, n}];

Array[a, 30, 0] (* Jean-Fran├žois Alcover, Jul 21 2018, after Vladimir Kruchinin *)

CoefficientList[Series[1/Sqrt[1-2*x-3*x^2-4*x^3], {x, 0, 30}], x] (* G. C. Greubel, Feb 18 2019 *)

PROG

(PARI) /* same as in A092566 but use */

steps=[[1, 0], [1, 1], [1, 2], [2, 1]];

/* Joerg Arndt, Jun 30 2011 */

(PARI) my(x='x+O('x^30)); Vec(1/sqrt(1-2*x-3*x^2-4*x^3)) \\ G. C. Greubel, Feb 18 2019

(Maxima)

a(n):=sum(binomial(2*k, k) * sum(binomial(j, n-k-j) * 2^(j-k) * binomial(k, j) * 3^(-n+k+2*j) * 4^(n-k-2*j), j, 0, k), k, 0, n); /* Vladimir Kruchinin, Feb 27 2016 */

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-2*x-3*x^2-4*x^3) )); // G. C. Greubel, Feb 18 2019

(Sage) (1/sqrt(1-2*x-3*x^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019

CROSSREFS

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369.

Sequence in context: A244826 A183111 A132835 * A001189 A212352 A198180

Adjacent sequences:  A191351 A191352 A191353 * A191355 A191356 A191357

KEYWORD

nonn

AUTHOR

Joerg Arndt, Jun 30 2011

STATUS

approved

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Last modified January 29 10:34 EST 2020. Contains 331337 sequences. (Running on oeis4.)