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A025564
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Triangular array, read by rows: pairwise sums of trinomial array A027907.
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15
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1, 1, 1, 1, 2, 1, 1, 3, 4, 3, 1, 1, 4, 8, 10, 8, 4, 1, 1, 5, 13, 22, 26, 22, 13, 5, 1, 1, 6, 19, 40, 61, 70, 61, 40, 19, 6, 1, 1, 7, 26, 65, 120, 171, 192, 171, 120, 65, 26, 7, 1, 1, 8, 34, 98, 211, 356, 483, 534, 483, 356, 211, 98, 34, 8, 1, 1, 9, 43, 140, 343, 665, 1050, 1373
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| T(n,k) is the number of strings of nonnegative integers "s(1)s(2)s(3)...s(k)" such that s(1)+s(2)+s(3)+...+s(k)=n and the string does not the substring "00". E.g. T(3,5) = 8 because the valid strings are 02010, 01020, 11010, 10110, 10101, 01110, 01101 and 01011. T(4,3) = 13, counting 040, 311, 301, 130, 031, 103, 013, 220, 202, 022, 211, 121 and 112 - Jose Luis Arregui (arregui(AT)unizar.es), Dec 05 2007
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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, 2, 1], [1, 3, 4, 3, 1].
G.f.: (1+yz)/[1-z(1+y+y^2)].
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EXAMPLE
| ..........1
.......1..1..1
....1..3..4..3..1
..1.4..8..10.8..4..1
1.5.13.22.26.22.13.5.1
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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, 2, 1][k+1], if(n==2, [1, 3, 4, 3, 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)))}
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CROSSREFS
| Columns include A025565, A025566, A025567, A025568.
Cf. A025177.
Sequence in context: A145034 A159933 A128314 * A052265 A055068 A015138
Adjacent sequences: A025561 A025562 A025563 * A025565 A025566 A025567
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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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