The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A317664 G.f.: Sum_{n>=0} ( (1+x)^n - 1 )^n * 4^n / (5 - 4*(1+x)^n)^(n+1). 4
 1, 4, 96, 3520, 181584, 12046208, 976817408, 93618157824, 10353263884352, 1297682198608960, 181792547403610112, 28148715766252519424, 4773717142486206475264, 879979421777903153737728, 175192929827140711780067328, 37462651348142346656294109184, 8563418069261195349710481467648, 2083773631690873034841394464054272 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The following identities hold for |y| <= 1 and fixed real k > 0: (C1) Sum_{n>=0} (y^n + k)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1). (C2) Sum_{n>=0} (y^n + 1)^n*k^n/(1+k + k*y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n*k^n/(1+k - k*y^n)^(n+1). This sequence is an example of (C2) when y = 1+x and k = 4. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..300 FORMULA G.f. A(x) satisfies: (1) A(x) = Sum_{n>=0} ( (1+x)^n - 1 )^n * 4^n / (5 - 4*(1+x)^n)^(n+1). (2) A(x) = Sum_{n>=0} ( (1+x)^n + 1 )^n * 4^n / (5 + 4*(1+x)^n)^(n+1). a(n) ~ c * d^n * n! / sqrt(n), where d = 14.74821884963947298733792887778672923688310694846410198271766770874395484... and c = 0.329067655604412806858767072708083473088299024445... - Vaclav Kotesovec, Aug 09 2018 EXAMPLE G.f.: A(x) = 1 + 4*x + 96*x^2 + 3520*x^3 + 181584*x^4 + 12046208*x^5 + 976817408*x^6 + 93618157824*x^7 + 10353263884352*x^8 + ... such that A(x) = 1 + ((1+x) - 1)*4/(5 - 4*(1+x))^2 + ((1+x)^2 - 1)^2*4^2/(5 - 4*(1+x)^2)^3 + ((1+x)^3 - 1)^3*4^3/(5 - 4*(1+x)^3)^4 + ((1+x)^4 - 1)^4*4^4/(5 - 4*(1+x)^4)^5 + ((1+x)^5 - 1)^5*4^5/(5 - 4*(1+x)^5)^6 + ((1+x)^6 - 1)^6*4^6/(5 - 4*(1+x)^6)^7 + ... Also, A(x) = 1/9 + ((1+x) + 1)*4/(5 + 4*(1+x))^2 + ((1+x)^2 + 1)^2*4^2/(5 + 4*(1+x)^2)^3 + ((1+x)^3 + 1)^3*4^3/(5 + 4*(1+x)^3)^4 + ((1+x)^4 + 1)^4*4^4/(5 + 4*(1+x)^4)^5 + ((1+x)^5 + 1)^5*4^5/(5 + 4*(1+x)^5)^6 + ((1+x)^6 + 1)^6*4^6/(5 + 4*(1+x)^6)^7 + ... PROG (PARI) {a(n) = my(A=1); A = sum(m=0, n, ( (1+x)^m - 1 +x*O(x^n) )^m * 4^m / (5 - 4*(1+x)^m +x*O(x^n) )^(m+1) ); ; polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A302598, A317662, A317663, A302615. Sequence in context: A206645 A267907 A221148 * A062779 A077155 A013042 Adjacent sequences: A317661 A317662 A317663 * A317665 A317666 A317667 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 03 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 29 10:31 EST 2022. Contains 358424 sequences. (Running on oeis4.)