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 A291813 G.f. A(x) satisfies: A(x - 2*x*A(x)) = x - x*A(x). 7
 1, 1, 5, 35, 297, 2873, 30657, 353727, 4355497, 56709337, 775575269, 11085971235, 164979882033, 2548461481105, 40762085472929, 673751263927071, 11489101983573105, 201838769635965969, 3648620371959258149, 67795012307507004291, 1293607920940368319641, 25326486746707799668105, 508368313083167614599201, 10454499119633293760277151, 220120546753823908307191769, 4742197866143368618862457641 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Table of n, a(n) for n=1..26. FORMULA G.f. A(x) also satisfies: (1) A(x) = (1/2)*Series_Reversion( x - 2*x*A(x) ) + x/2. (2) A( 2*A(x) - x ) = (A(x) - x) / (2*A(x) - x). a(n) = Sum_{k=0..n-1} A291820(n, k) * 2^k. EXAMPLE G.f.: A(x) = x + x^2 + 5*x^3 + 35*x^4 + 297*x^5 + 2873*x^6 + 30657*x^7 + 353727*x^8 + 4355497*x^9 + 56709337*x^10 + 775575269*x^11 + 11085971235*x^12 +... such that A(x - 2*x*A(x)) = x - x*A(x). RELATED SERIES. A(x - 2*x*A(x)) = x - x^2 - x^3 - 5*x^4 - 35*x^5 - 297*x^6 - 2873*x^7 - 30657*x^8 +... which equals x - x*A(x). Series_Reversion( x - 2*x*A(x) ) = x + 2*x^2 + 10*x^3 + 70*x^4 + 594*x^5 + 5746*x^6 + 61314*x^7 + 707454*x^8 + 8710994*x^9 + 113418674*x^10 +... which equals 2*A(x) - x. A( 2*A(x) - x ) = x + 3*x^2 + 19*x^3 + 159*x^4 + 1561*x^5 + 17087*x^6 + 202975*x^7 + 2574391*x^8 + 34495545*x^9 + 484770627*x^10 + 7107406323*x^11 + 108289787415*x^12 + 1709478736593*x^13 + 27894511442079*x^14 +... which equals (A(x) - x) / (2*A(x) - x). PROG (PARI) {a(n) = my(A=x); for(i=1, n, A = (1/2)*serreverse( x - 2*x*A +x*O(x^n) ) + x/2 ); polcoeff(A, n)} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A291820, A291814, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744. Sequence in context: A051406 A000356 A027392 * A346765 A177354 A253096 Adjacent sequences: A291810 A291811 A291812 * A291814 A291815 A291816 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 01 2017 STATUS approved

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Last modified February 21 07:26 EST 2024. Contains 370219 sequences. (Running on oeis4.)