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A025177
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Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.
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23
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1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 4, 11, 20, 29, 32, 29, 20, 11, 4, 1, 1, 5, 16, 35, 60, 81, 90, 81, 60, 35, 16, 5, 1, 1, 6, 22, 56, 111, 176, 231, 252, 231, 176, 111, 56, 22, 6, 1, 1, 7, 29, 84, 189, 343, 518, 659, 714, 659, 518, 343
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OFFSET
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0,7
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COMMENTS
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The Motzkin transforms of the rows starting (1, 2), (1, 3) and (1, 4), extended by zeros after their last element, are apparently in A026134, A026109 and A026110. - R. J. Mathar, Dec 11 2008
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LINKS
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FORMULA
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T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1].
G.f.: (1-y*z)/[1-z*(1+y+y^2)].
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EXAMPLE
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1
1 0 1
1 1 2 1 1
1 2 4 4 4 2 1
1 3 7 10 12 10 7 3 1
1 4 11 20 29 32 29 20 11 4 1
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MAPLE
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option remember;
if k < 0 or k > 2*n then
0;
elif n = 0 then
1 ;
elif n = 1 then
op(k+1, [1, 0, 1]) ;
else
procname(n-1, k-2)+procname(n-1, k-1)+procname(n-1, k) ;
end if;
end proc:
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MATHEMATICA
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nmax = 10; CoefficientList[CoefficientList[Series[(1 - y*x)/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)
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PROG
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(PARI) {T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1, 0, 1][k+1], if( n==2, [1, 1, 2, 1, 1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};
(PARI) T(n, k)=polcoeff(Ser(polcoeff(Ser((1-y*z)/(1-z*(1+y+y^2)), y), k, y), z), n, z)
(PARI) {T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, polcoeff( (1 + x + x^2)^n, k) - polcoeff( (1 + x + x^2)^(n-1), k-1)))};
(PARI) g=matrix(33, 65);
for(n=0, 32, for(k=0, 2*n, g[n+1, k+1]=0));
g[1, 1]=1;
g[2, 1]=1; g[2, 2]=0; g[2, 3]=1;
g[3, 1]=1; g[3, 2]=1; g[3, 3]=2; g[3, 4]=1; g[3, 5]=1;
for(n=0, 2, for(k=0, 2*n, print(n, " ", k, " ", g[n+1, k+1])))
for(n=3, 32, g[n+1, 1]=1; print(n, " 1 1"); g[n+1, 2]=n-1; print(n, " 2 ", n-1); for(k=2, 2*n, g[n+1, k+1]=g[n, k-1]+g[n, k]+g[n, k+1]; print(n, " ", k, " ", g[n+1, k+1])))
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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