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A105632 Triangle, read by rows, where the g.f. A(x,y) satisfies the equation: A(x,y) = 1/(1-x*y) + x*A(x,y) + x^2*A(x,y)^2. 4
1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 7, 4, 1, 1, 21, 19, 10, 5, 1, 1, 51, 51, 31, 13, 6, 1, 1, 127, 141, 91, 45, 16, 7, 1, 1, 323, 393, 276, 141, 61, 19, 8, 1, 1, 835, 1107, 834, 461, 201, 79, 22, 9, 1, 1, 2188, 3139, 2535, 1485, 701, 271, 99, 25, 10, 1, 1, 5798, 8953, 7711, 4803 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Column 0 is A001006 (Motzkin numbers). Column 1 is A002426 (Central trinomial coefficients). Row sums form A105633 (also equal to A057580?).

The matrix inverse starts

1;

-1,1;

-1,-1,1;

0,-2,-1,1;

2,-1,-3,-1,1;

6,2,-2,-4,-1,1;

13,10,2,-3,-5,-1,1;

18,32,14,2,-4,-6,-1,1;

-12,76,56,18,2,-5,-7,-1,1;

-206,108,162,86,22,2,-6,-8,-1,1;

- R. J. Mathar, Apr 08 2013

LINKS

Table of n, a(n) for n=0..69.

FORMULA

G.f. for column k (k>0): Sum_{j=0..k-1} C(k-1, j)*A000108(j)*x^(2*j)/(1-2*x-3*x^2)^(j+1/2), where A000108(j) = binomial(2*j, j)/(j+1) is the j-th Catalan number.

G.f.: A(x, y) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x*y)))/(2*x^2).

EXAMPLE

Triangle begins:

1;

1,1;

2,1,1;

4,3,1,1;

9,7,4,1,1;

21,19,10,5,1,1;

51,51,31,13,6,1,1;

127,141,91,45,16,7,1,1;

323,393,276,141,61,19,8,1,1;

835,1107,834,461,201,79,22,9,1,1; ...

Let G = (1-2*x-3*x^2), then the column g.f.s are:

k=1: 1/sqrt(G)

k=2: (G +(1)*1*x^2)/sqrt(G^3)

k=3: (G^2 +(1)*2*x^2*G +(2)*1*x^4)/sqrt(G^5)

k=4: (G^3 +(1)*3*x^2*G^2 +(2)*3*x^4*G +(5)*1*x^6)/sqrt(G^7)

k=5: (G^4 +(1)*4*x^2*G^3 +(2)*6*x^4*G^2 +(5)*4*x^6*G +(14)*1*x^8)/sqrt(G^9)

and involve Catalan numbers and binomial coefficients.

MAPLE

A105632 := proc(n, k)

    (1-x-sqrt((1-x)^2-4*x^2/(1-x*y)))/2/x^2 ;

    coeftayl(%, x=0, n) ;

    coeftayl(%, y=0, k) ;

end proc: # R. J. Mathar, Apr 08 2013

PROG

(PARI) {T(n, k)=local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1, n, A=1/(1-x*y)+x*A+x^2*A^2); polcoeff(polcoeff(A, n, x), k, y)}

(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff( 2/(1-X+sqrt((1-X)^2-4*X^2/(1-X*Y)))/(1-X*Y), n, x), k, y)}

CROSSREFS

Cf. A105633 (row sums), A001006 (column 0), A002426 (column 1).

Sequence in context: A217781 A339428 A204849 * A091491 A117418 A101494

Adjacent sequences:  A105629 A105630 A105631 * A105633 A105634 A105635

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 17 2005

STATUS

approved

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Last modified July 24 04:25 EDT 2021. Contains 346273 sequences. (Running on oeis4.)