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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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4
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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
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OFFSET
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0,4
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COMMENTS
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Row sums = A027307 (paths from (0,0) to (3n,0) in steps (2,1),(1,2),(1,-1)).
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LINKS
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Table of n, a(n) for n=0..45.
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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).
Antidiagonal sums = A001002 (number of dissections of a polygon).
Semidiagonal sums = A104979.
G.f.: A(x,y) = Sum_{n>=0} x^n/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
G.f. of row n: 1/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
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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;
1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263; ...
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PROG
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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)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) {Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D, y)); D}
{T(n, k)=local(A=1); A=1+sum(m=1, n+1, x^m/y^(m+1) * Dy(m-1, (y^2+y^3)^m/m!)) +x*O(x^n)+y*O(y^k); polcoeff(polcoeff(A, n, x), k, y)} \\ Paul D. Hanna, Jun 22 2012
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
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CROSSREFS
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Cf. A000108, A001764, A027307, A001002, A104979.
Sequence in context: A224166 A216052 A021911 * A124568 A091807 A085825
Adjacent sequences: A104975 A104976 A104977 * 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, Mar 30 2005
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STATUS
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approved
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