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 A120021 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-(n+1)*x) o (x-x^2) } for n>=1. 3
 1, 3, 16, 130, 1416, 19236, 312512, 5906502, 127313320, 3082645951, 82848394752, 2447576485341, 78846484722208, 2750891289611235, 103344880800464896, 4159577854374314795, 178587276548655542112, 8147334149686335230068 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Secondary diagonal of A120019, the table of self-compositions of A120010. LINKS FORMULA a(n) = Sum_{j=1..n} Catalan(n-j)*[Sum_{i=1..j} (-1)^(j-i)*(n+1)^(i-1)*C(n-j+i, j-i)*C(n-j+i-1, i-1)], where Catalan(n) = A000108(n) = C(2n, n)/(n+1). EXAMPLE Successive self-compositions of F(x), the g.f. of A120010, begin: F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +... F(F(x)) = (1)x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +... F(F(F(x))) = x + (3)x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +... F(F(F(F(x)))) = x + 4x^2 + (16)x^3 + 68x^4 + 312x^5 + 1536x^6 +... F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + (130)x^4 + 710x^5 + 4070x^6 +... F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + (1416)x^5 + 9348x^6+.. PROG (PARI) a(n)=polcoeff((1-sqrt(1-4*x*(1-x)/(1-(n+1)*x*(1-x)+x*O(x^n))))/2, n, x) (PARI) /* Alternative Formula: */ a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)*sum(i=1, j, (-1)^(j-i)*(n+1)^(i-1)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1))) CROSSREFS Cf. A120010, A120019, A120020. Sequence in context: A340341 A135752 A218827 * A223897 A131490 A121673 Adjacent sequences:  A120018 A120019 A120020 * A120022 A120023 A120024 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 14 2006 STATUS approved

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Last modified August 2 17:49 EDT 2021. Contains 346428 sequences. (Running on oeis4.)