OFFSET
0,7
COMMENTS
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
LINKS
Georg Fischer, Table of n, a(n) for n = 0..2499 [rows 0..49; first 676 terms from G. C. Greubel]
FORMULA
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)].
EXAMPLE
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
MAPLE
A025177 := proc(n, k)
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:
seq(seq(A025177(n, k), k=0..2*n), n=0..20) ; # R. J. Mathar, Feb 25 2015
MATHEMATICA
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 *)
PROG
(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])))
\\ Michael B. Porter, Feb 02 2010
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
EXTENSIONS
Edited by Ralf Stephan, Jan 09 2005
Offset corrected by R. J. Mathar, Feb 25 2015
STATUS
approved