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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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22
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,7
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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. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 11 2008]
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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-yz)/[1-z(1+y+y^2)].
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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
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PROG
| (PARI) T(n, k)=if(n<0||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(n<0||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)))}
Contribution from Michael B. Porter (michael_b_porter(AT)yahoo.com), Feb 02 2010: (Start)
(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]))) (End)
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CROSSREFS
| Columns include A025178, A025179, A025180, A025181, A025182.
Cf. A024996.
Sequence in context: A145515 A188919 A026519 * A026148 A117211 A061545
Adjacent sequences: A025174 A025175 A025176 * A025178 A025179 A025180
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KEYWORD
| nonn,tabf,easy
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Edited by Ralf Stephan, Jan 09 2005
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