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 A258313 G.f. A(x) satisfies: A(x) = B(x)*C(x) where B(x) = 1 + x*A(x)*C(x) and C(x) = 1 + 2*x*A(x)*B(x). 3
 1, 3, 15, 93, 641, 4719, 36335, 289017, 2356321, 19586283, 165364799, 1414193205, 12224831937, 106645825047, 937685498271, 8301129707121, 73929906605249, 661919872559763, 5954449287679919, 53791836313371405, 487807821246726273, 4438980860105747967, 40521481906592540175 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA G.f. A(x) satisfies: (1) A(x) = 1 + 3*x*A(x) + 2*x^2*A(x)^2*(1 + 2*A(x)) - 4*x^4*A(x)^5. (2) A(x) = (1 + x*A(x))*(1 + 2*x*A(x)) / (1 - 2*x^2*A(x)^2)^2. (3) A(x) = (1/x) * Series_Reversion( x*(1-2*x^2)^2 / ((1+x)*(1+2*x)) ). Other relations involving A=A(x), B=B(x), and C=C(x) are: (a) B = (1 + x*A) / (1 - 2*x^2*A^2). (b) C = (1 + 2*x*A) / (1 - 2*x^2*A^2). (c) B = 1/(1 - x*C^2). (d) C = 1/(1 - 2*x*B^2). a(n) = Sum_{k=0..n} 2^k * (2*n+1)/((2*n-2*k+1)*(2*k+1)) * binomial(2*n-k, k) * binomial(n+k, n-k). - Paul D. Hanna, Dec 08 2016 Recurrence: 16*n*(n+1)*(2*n-1)*(2*n+1)*(78144*n^5 - 638176*n^4 + 2009556*n^3 - 3030476*n^2 + 2162967*n - 571095)*a(n) = 6*n*(2*n - 1)*(3750912*n^7 - 30632448*n^6 + 95859584*n^5 - 141041184*n^4 + 91266236*n^3 - 10305348*n^2 - 11143087*n + 2769495)*a(n-1) + 18*(1875456*n^9 - 19067136*n^8 + 81388448*n^7 - 188788320*n^6 + 255050924*n^5 - 194874764*n^4 + 66686587*n^3 + 7734535*n^2 - 12646725*n + 2646000)*a(n-2) + 18*(n-2)*(1875456*n^8 - 18129408*n^7 + 70578528*n^6 - 140304800*n^5 + 146662564*n^4 - 69042202*n^3 - 184198*n^2 + 11212005*n - 2646000)*a(n-3) + 3*(n-3)*(n-2)*(3*n - 10)*(3*n - 5)*(78144*n^5 - 247456*n^4 + 238292*n^3 - 49424*n^2 - 31301*n + 10920)*a(n-4). - Vaclav Kotesovec, Dec 09 2016 EXAMPLE G.f.: A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 641*x^4 + 4719*x^5 + 36335*x^6 +... where A(x) = B(x)*C(x): B(x) = 1 + x + 5*x^2 + 29*x^3 + 193*x^4 + 1389*x^5 + 10525*x^6 +... C(x) = 1 + 2*x + 8*x^2 + 46*x^3 + 304*x^4 + 2178*x^5 + 16456*x^6 +... Related series: A(x)*B(x) = 1 + 4*x + 23*x^2 + 152*x^3 + 1089*x^4 + 8228*x^5 +... A(x)*C(x) = 1 + 5*x + 29*x^2 + 193*x^3 + 1389*x^4 + 10525*x^5 +... MATHEMATICA Table[Sum[2^k*(2*n + 1)/((2*n - 2*k + 1)*(2*k + 1))*Binomial[2*n - k, k]*Binomial[n + k, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 09 2016, after Paul D. Hanna *) PROG (PARI) {a(n)=local(A=1+x, B=1+x, C=1+2*x); for(i=1, n, A = B*C +x*O(x^n); B = 1 + x*A*C + x*O(x^n); C = 1 + 2*x*A*B + x*O(x^n)); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=local(A=1); A = (1/x) * serreverse( x*(1-2*x^2)^2 / ((1+x)*(1+2*x) +x*O(x^n)) ); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) /* Explicit formula from triangle A278880 */ {a(n) = sum(k=0, n, 2^k * (2*n+1)/((2*n-2*k+1)*(2*k+1)) * binomial(2*n-k, k) * binomial(n+k, n-k) )} for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 08 2016 CROSSREFS Cf. A258314 (B(x)), A258315 (C(x)), A278880. Sequence in context: A192296 A002893 A256335 * A074539 A103210 A203014 Adjacent sequences:  A258310 A258311 A258312 * A258314 A258315 A258316 KEYWORD nonn AUTHOR Paul D. Hanna, May 25 2015 STATUS approved

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Last modified January 19 00:40 EST 2020. Contains 331030 sequences. (Running on oeis4.)