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Search: id:A104978
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| A104978 |
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Triangle where g.f. satisfies: A(x,y) = 1 + x*A(x,y)^2 + x*y*A(x,y)^3, read by rows. |
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+0
5
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| 1, 1, 1, 2, 5, 3, 5, 21, 28, 12, 14, 84, 180, 165, 55, 42, 330, 990, 1430, 1001, 273, 132, 1287, 5005, 10010, 10920, 6188, 1428, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263, 4862
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Row sums = A027307 (paths from (0,0) to (3n,0) in steps (2,1),(1,2),(1,-1)). Antidiagonal sums = A001002 (number of dissections of a polygon). Semidiagonal sums = A104979.
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FORMULA
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T(n, k) = binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1). Column 0: T(n, 0) = A000108(n) (Catalan numbers). Main diagonal: T(n, n) = A001764(n) (ternary tree numbers).
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EXAMPLE
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Triangle begins:
1;
1,1;
2,5,3;
5,21,28,12;
14,84,180,165,55;
42,330,990,1430,1001,273;
132,1287,5005,10010,10920,6188,1428;
429,5005,24024,61880,92820,81396,38760,7752; ...
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PROGRAM
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(PARI) {T(n, k)=local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1, n, A=1+x*A^2+x*y*A^3); polcoeff(polcoeff(A, n, x), k, y)}
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CROSSREFS
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Cf. A000108, A001764, A027307, A001002, A104979.
Sequence in context: A026207 A078376 A021911 this_sequence A124568 A091807 A085825
Adjacent sequences: A104975 A104976 A104977 this_sequence A104979 A104980 A104981
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005
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