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 A213591 G.f. A(x) satisfies: A( x - A(x)^2 ) = x. 32
 1, 1, 4, 24, 178, 1512, 14152, 142705, 1528212, 17211564, 202460400, 2474708496, 31310415376, 408815254832, 5495451727376, 75907303147652, 1075685334980240, 15618612118252960, 232102241507321384, 3526880759915999016, 54755450619399484512, 867928449982022915984 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Unsigned version of A139702. Self-convolution is A276370. Row sums of triangle A277295. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 1..300 FORMULA G.f. satisfies: (1) A(x) = x + A(A(x))^2. (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) / n!. (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ). (4) A(x) = x*G(A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411. (5) A(x)^2 = x*F(A(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628. (6) x = A(A( x-x^2 - A(x)^2 )). - Paul D. Hanna, Jul 01 2012 (7) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with: A = x + B^2; B = A + C^2; C = B + D^2; D = C + E^2;  ... where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc. ... a(n) = Sum_{k=0..n-1} A277295(n,k). EXAMPLE G.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +... where A(x) = x + A(A(x))^2: A(A(x)) = x + 2*x^2 + 10*x^3 + 69*x^4 + 568*x^5 + 5250*x^6 + 52792*x^7 +... A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +... The g.f. satisfies the series: A(x) = x + A(x)^2 + d/dx A(x)^4/2! + d^2/dx^2 A(x)^6/3! + d^3/dx^3 A(x)^8/4! +... Logarithmic series: log(A(x)/x) = A(x)^2/x + [d/dx A(x)^4/x]/2! + [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! +... Also, A(x) = x*G(A(x)^2/x) where G(x) = x/A(x/G(x)^2) is the g.f. of A212411: G(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 + 15261*x^7 +... Also, A(x)^2 = x*F(A(x)) where F(x) is the g.f. of A213628: F(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 + 46013*x^8 +... MATHEMATICA terms = 22; A[_] = 0; Do[A[x_] = x + A[A[x]]^2 + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 09 2018 *) PROG (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^2+x*O(x^n))); polcoeff(A, n))} (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)} (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)} for(n=1, 21, print1(a(n), ", ")) CROSSREFS Cf. A220379, A139702, A212411, A138740, A213628, A213639. Cf. A088714, A088717, A091713, A120971, A140094, A140095. Cf. A143426, A087949, A143435, A182969. Cf. A275765, A276360, A276361, A276362, A276363, A276370. Cf. A277295 (triangle). Sequence in context: A007846 A337507 A139702 * A243689 A309637 A168452 Adjacent sequences:  A213588 A213589 A213590 * A213592 A213593 A213594 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 14 2012 STATUS approved

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Last modified September 17 09:00 EDT 2021. Contains 347478 sequences. (Running on oeis4.)