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 A120406 Triangle read by rows: related to series expansion of the square root of 2 linear factors. 2
 1, 2, 2, 5, 6, 5, 14, 18, 18, 14, 42, 56, 60, 56, 42, 132, 180, 200, 200, 180, 132, 429, 594, 675, 700, 675, 594, 429, 1430, 2002, 2310, 2450, 2450, 2310, 2002, 1430, 4862, 6864, 8008, 8624, 8820, 8624, 8008, 6864, 4862 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The numbers T(n,k) arise in the expansion of the square root of 2 generic linear factors: 1 - sqrt((1-a*x)*(1-b*x)) = (a+b)*x/2 + (1/8)*(b-a)^2*x^2*Sum_{n>=0} (Sum_{k=0..n} T(n,k)*a^k*b^(n-k))*(x/4)^n. (The g.f. below simply reformulates this fact.) A combinatorial interpretation of T(n,k) would be very interesting. LINKS FORMULA T(n,k) = 2*binomial(n,k)^2*binomial(2n+2,n)/binomial(2n+2,2k+1). This shows that T(n,k) is positive and the rows are symmetric. T(n,k) = (k+1)*CatalanNumber(n+1) - 2*Sum_{j=0..k-1} (k-j)*CatalanNumber(j)*CatalanNumber(n-j). This shows that T(n,k) is an integer. Generating function F(x,y):=Sum_{n>=0, k=0..n} T(n,k) x^n y^k is given by F(x,y) = ( 1-2x-2x*y-sqrt(1-4x)*sqrt(1-4x*y) )/( 2x^2*(1-y)^2 ). This shows that the row sums are the powers of 4 (A000302) because lim_{y->1} F(x,y) = 1/(1-4x). 1 + x*(d/dx)(log(F(x,y))) = 1 + (2 + 2*y)*x + (6 + 4*y + 6*y^2)*x^2 + ... is the o.g.f. for A067804. - Peter Bala, Jul 17 2015 G.f. A(x,y) = -G(-x,y), G(x,y) satisfies G(x,y) = x/A008459(G(x,y))^2. - Vladimir Kruchinin, Oct 24 2020 EXAMPLE Table begins \ k..0....1....2....3....4....5....6 n 0 |..1 1 |..2....2 2 |..5....6....5 3 |.14...18...18...14 4 |.42...56...60...56...42 5 |132..180..200..200..180..132 6 |429..594..675..700..675..594..429 MATHEMATICA Table[2 Binomial[n, k]^2 Binomial[2n+2, n]/ Binomial[2n+2, 2k+1], {n, 0, 9}, {k, 0, n}] PROG (Maxima) solve(A=x*(A^2*y^2-2*A^2*y-2*A*y+A^2-2*A+1), A); /* Vladimir Kruchinin, oct 24 2020 */ CROSSREFS Column k=0 is the Catalan numbers A000108 (offset). The middle-of-row entries form A005566. Cf. A067804. Sequence in context: A112573 A233740 A266595 * A050157 A209503 A209744 Adjacent sequences:  A120403 A120404 A120405 * A120407 A120408 A120409 KEYWORD nonn,tabl AUTHOR David Callan, Jul 03 2006 STATUS approved

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Last modified December 3 13:43 EST 2021. Contains 349463 sequences. (Running on oeis4.)