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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 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: A033185 A217781 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 December 3 12:20 EST 2020. Contains 338902 sequences. (Running on oeis4.)