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 A199248 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^k]* x^n/n ), where A027907 is the triangle of trinomial coefficients. 3
 1, 1, 2, 6, 20, 69, 248, 923, 3523, 13706, 54152, 216710, 876607, 3578405, 14722432, 60986158, 254145337, 1064712328, 4481577078, 18943753140, 80381689202, 342254333393, 1461864544896, 6262021627055, 26894816382199, 115792035533779, 499648608539714, 2160504474956390 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n. LINKS FORMULA G.f. satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2. G.f.: A(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12) ). EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 248*x^6 + 923*x^7 +... such that A(x) = G(x*A(x)) where G(x) is given by: G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2 = (1-x^5)/(1-x) + x^3/(1-x)^2: G(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 7*x^9 +... ... Let A = x*A(x), then the logarithm of the g.f. A(x) equals the series: log(A(x)) = (1 + A + A^2)*x + (1 + 2^2*A + 3^2*A^2 + 2^2*A^3 + A^4)*x^2/2 + (1 + 3^2*A + 6^2*A^2 + 7^2*A^3 + 6^2*A^4 + 3^2*A^5 + A^6)*x^3/3 + (1 + 4^2*A + 10^2*A^2 + 16^2*A^3 + 19^2*A^4 + 16^2*A^5 + 10^2*A^6 + 4^2*A^7 + A^8)*x^4/4 + (1 + 5^2*A + 15^2*A^2 + 30^2*A^3 + 45^2*A^4 + 51^2*A^5 + 45^2*A^6 + 30^2*A^7 + 15^2*A^8 + 5^2*A^9 + A^10)*x^5/5 +... which involves the squares of the trinomial coefficients A027907(n,k). PROG (PARI) {a(n)=local(A=1+x); A=1/x*serreverse(x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12+x*O(x^n))); polcoeff(A, n)} (PARI) /* G.f. A(x) using the squares of the trinomial coefficients */ {A027907(n, k)=polcoeff((1+x+x^2)^n, k)} {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k*A^k) *x^m/m)+x*O(x^n))); polcoeff(A, n)} CROSSREFS Cf. A186236, A199257, A027907. Sequence in context: A163135 A331951 A047036 * A148478 A148479 A150124 Adjacent sequences:  A199245 A199246 A199247 * A199249 A199250 A199251 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 04 2011 STATUS approved

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Last modified December 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)