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%I #25 Sep 24 2019 12:15:58
%S 1,4,16,78,404,2208,12492,72589,430569,2596471,15870357,98102191,
%T 612222083,3852015239,24408653703,155629858911,997744376239,
%U 6427757480074,41590254520410,270163621543421,1761179219680657
%N Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 3.
%C Column 3 of A204213
%C 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
%H R. H. Hardin, <a href="/A204208/b204208.txt">Table of n, a(n) for n = 1..210</a>
%H C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, <a href="https://arxiv.org/abs/1609.06473">Explicit formulas for enumeration of lattice paths: basketball and the kernel method</a>, arXiv preprint arXiv:1609.06473 [math.CO], 2016.
%F 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
%F 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
%e Some solutions for n=5
%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e ..2....1....3....3....2....2....1....2....0....0....2....3....0....3....1....2
%e ..5....3....2....2....2....3....1....5....3....0....2....4....3....2....0....3
%e ..2....6....3....4....0....1....0....6....5....1....0....6....5....2....2....5
%e ..2....3....3....3....2....3....3....3....2....1....0....3....3....0....3....3
%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%t 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];
%t a /@ Range[1, 21] (* _Jean-François Alcover_, Sep 24 2019, after _Vladimir Kruchinin_ *)
%o (PARI) {A025012(n)=polcoeff((1+x+x^2+x^3+x^4+x^5+x^6 +x*O(x^(3*n)))^n,3*n)}
%o {a(n)=polcoeff(exp(sum(m=1,n,A025012(m)*x^m/m)+x*O(x^n)),n)}
%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 01 2013
%o (Maxima)
%o 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 */
%K nonn
%O 1,2
%A _R. H. Hardin_, Jan 12 2012
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