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 A193037 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^A022998(n), where A022998 is defined as "if n is odd then n else 2*n.". 4
 1, 1, 3, 16, 99, 660, 4625, 33609, 251024, 1915365, 14866307, 117007587, 931682106, 7491746385, 60750081839, 496214311987, 4078991375519, 33718664525501, 280123674031062, 2337556609209193, 19584517345276853, 164677962557101656, 1389268739557153255 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare the g.f. to a g.f. C(x) of the Catalan numbers: x = Sum_{n>=1} x^n*C(-x)^(2*n-1). LINKS FORMULA G.f. satisfies: 1 = A(x)/(1 - x^2*A(x)^2) - x*A(x)^4/(1 - x^2*A(x)^4). G.f. satisfies: A(x) = 1 - x^2*A(x)^2 + x*(1-x)*A(x)^4 + x^2*A(x)^5 - x^3*(1-x)*A(x)^6. EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 99*x^4 + 660*x^5 + 4625*x^6 +... The g.f. satisfies: x = x*A(-x) + x^2*A(-x)^4 + x^3*A(-x)^3 + x^4*A(-x)^8 + x^5*A(-x)^5 + x^6*A(-x)^12 +...+ x^n*A(-x)^A022998(n) +... where A022998 begins: [1,4,3,8,5,12,7,16,9,20,11,24,13,28,15,32,...]. PROG (PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^((2-m%2)*m)), #A)); if(n<0, 0, A[n+1])} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^4-x^2*(A^2+A^4-A^5)-x^3*A^6+x^4*A^6+x*O(x^n)); polcoeff(A, n)} CROSSREFS Cf. A193036, A193038, A193039, A193040, A022998. Sequence in context: A233203 A074553 A303831 * A246056 A091641 A137572 Adjacent sequences:  A193034 A193035 A193036 * A193038 A193039 A193040 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 14 2011 STATUS approved

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Last modified February 18 02:57 EST 2020. Contains 332006 sequences. (Running on oeis4.)