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A204208 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 3. 2
1, 4, 16, 78, 404, 2208, 12492, 72589, 430569, 2596471, 15870357, 98102191, 612222083, 3852015239, 24408653703, 155629858911, 997744376239, 6427757480074, 41590254520410, 270163621543421, 1761179219680657 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column 3 of A204213

Number of excursions (walks starting at the origin, ending on the x-axis, and never go below the x-axis in between) with n steps from {-3,-2,-1,0,1,2,3}. - David Nguyen, Dec 16 2016

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

FORMULA

G.f.: exp( Sum_{n>=1} A025012(n)*x^n/n ) - 1, where A025012(n) = central coefficient of (1+x+x^2+x^3+x^4+x^5+x^6)^n. - Paul D. Hanna, Aug 01 2013

a(n) = Sum_{i=1..n}((Sum_{j=0..(3*i)/7}(binomial(i,j)*binomial(-7*j+4*i-1,3*i-7*j)*(-1)^j))*a(n-i))/n. - Vladimir Kruchinin, Apr 06 2017

EXAMPLE

Some solutions for n=5

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

..2....1....3....3....2....2....1....2....0....0....2....3....0....3....1....2

..5....3....2....2....2....3....1....5....3....0....2....4....3....2....0....3

..2....6....3....4....0....1....0....6....5....1....0....6....5....2....2....5

..2....3....3....3....2....3....3....3....2....1....0....3....3....0....3....3

..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

MATHEMATICA

a[n_] := a[n] = If[n == 0, 1, Sum[(Sum[Binomial[i, j] Binomial[-7j + 4i - 1, 3i - 7j] (-1)^j, {j, 0, (3i)/7}]) a[n - i], {i, 1, n}]/n];

a /@ Range[1, 21] (* Jean-Fran├žois Alcover, Sep 24 2019, after Vladimir Kruchinin *)

PROG

(PARI) {A025012(n)=polcoeff((1+x+x^2+x^3+x^4+x^5+x^6 +x*O(x^(3*n)))^n, 3*n)}

{a(n)=polcoeff(exp(sum(m=1, n, A025012(m)*x^m/m)+x*O(x^n)), n)}

for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 01 2013

(Maxima)

a(n):=if n=0 then 1 else sum((sum(binomial(i, j)*binomial(-7*j+4*i-1, 3*i-7*j)*(-1)^j, j, 0, (3*i)/7))*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin, Apr 06 2017 */

CROSSREFS

Sequence in context: A020051 A020006 A207653 * A138294 A014514 A000780

Adjacent sequences:  A204205 A204206 A204207 * A204209 A204210 A204211

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 12 2012

STATUS

approved

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)