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 A145345 G.f. satisfies: A(x/A(x)) = 1 + x*A(x). 4
 1, 1, 2, 7, 34, 203, 1398, 10706, 89120, 794347, 7502170, 74511150, 773864654, 8368430208, 93905460014, 1090519614152, 13077315637592, 161643281777801, 2056306418177832, 26887064722265250, 360939404438509866 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Paul D. Hanna, Nov 15 2008: (Start) More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x); thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End) LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..200 FORMULA G.f. satisfies: A(x) = 1 + x*G(x)^2 where G(x) = g.f. of A121687. G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = g.f. of A121687. - Paul D. Hanna, Nov 08 2008 a(n) = [x^(n-1)] (2/(n+1))*A(x)^(n+1) for n>=1 with a(0)=1; i.e., a(n) equals 2/(n+1) times the coefficient of x^(n-1) in A(x)^(n+1) for n>=1. - Paul D. Hanna, Nov 15 2008 EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + ... A(x/A(x)) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 + ... A(x) = 1 + x*G(x)^2 where G(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + ... is the g.f. of A121687. ALTERNATE GENERATING METHOD. This sequence forms column zero in the following array. Let A denote this sequence. Start in row zero with this sequence, A, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A and the remaining terms in a given row to obtain the next row: [1, 1, 1, 2, 7, 34, 203, 1398, 10706, 89120, 794347, 7502170, ...]; [1, 2, 5, 18, 86, 502, 3387, 25496, 209242, 1843134, 17235671, ...]; [2, 7, 27, 128, 727, 4763, 34912, 280006, 2418537, 22240055, ...]; [7, 34, 169, 958, 6173, 44364, 349152, 2965098, 26864357, ...]; [34, 203, 1195, 7707, 54792, 425216, 3560600, 31842929, ...]; [203, 1398, 9308, 66310, 510689, 4231188, 37425922, ...]; [1398, 10706, 78414, 605401, 4987185, 43742924, 406387957, ...]; [10706, 89120, 705227, 5824356, 50853813, 469182452, ...]; [89120, 794347, 6707823, 58712463, 539651646, 5211277285, ...]; [794347, 7502170, 67008980, 617340184, 5942316416, 59827126712, ...]; ... PROG (PARI) {a(n)=local(F=1+x); for(i=0, n, G=serreverse(x/(F+x*O(x^n)))/x; F=1+x*subst(F, x, x*G)^2); polcoeff(F, n)} (PARI) {a(n)=local(F=1+x); for(i=0, n, G=serreverse(x/(F+x*O(x^n)))/x; F=1+x*G^2); polcoeff(F, n)} \\ Paul D. Hanna, Nov 08 2008 (PARI) /* This sequence is generated when k=1, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n, k=1, m=1)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))} \\ Paul D. Hanna, Nov 15 2008 (PARI) /* Prints terms 0..30 */ {A=[1]; for(m=1, 30, B=A; for(i=1, m-1, C=Vec(Ser(A)*Ser(B)); B=vector(#C-1, n, C[n+1]) ); A=concat(A, 0); A[#A]=B[1] ); A} \\ Paul D. Hanna, Jan 10 2016 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+1)))[m+1] - Vec(Ser(A))[m+1])/(m+1)); A[n+1]} for(n=0, 30, print1(2^n*a(n), ", ")) \\ Vaclav Kotesovec, Jan 31 2023 CROSSREFS Cf. A121687, A145350, A145349, A147664, A302703. Sequence in context: A307696 A237645 A117399 * A212027 A056543 A357829 Adjacent sequences: A145342 A145343 A145344 * A145346 A145347 A145348 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 05 2008 STATUS approved

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Last modified December 5 10:56 EST 2023. Contains 367589 sequences. (Running on oeis4.)