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 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A136654 G.f.: A(x) = (1/x)*Series_Reversion( x/Sum_{k=0..n} 2^[k(k-1)/2]*x^k ). 2
 1, 1, 3, 15, 117, 1565, 41663, 2378147, 286991465, 71261033273, 35889915535835, 36421251158141399, 74222529448186797341, 303194457634544530959125, 2480120130065258782050157847, 40601998283406419045206334661611 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..15. FORMULA a(n) = coefficient of x^n in [Sum_{k=0..n} 2^(k(k-1)/2)*x^k]^(n+1)/(n+1). EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 117*x^4 + 1565*x^5 + 41663*x^6 +... Let F(x) = 1 + x + 2x^2 + 8x^3 + 64x^4 + 1024x^5 +...+ 2^(n(n-1)/2)*x^n +.. then A(x) = F(x*A(x)), A(x/F(x)) = F(x). a(n) = coefficient of x^n in F(x)^(n+1)/(n+1), as can be seen by the main diagonal in the array of coefficients in the initial powers of F(x): F^1: [(1), 1, 2, 8, 64, 1024, 32768, 2097152, 268435456,...; F^2: [1, (2), 5, 20, 148, 2208, 67904, 4264960, 541216768,...; F^3: [1, 3, (9), 37, 258, 3588, 105704, 6507552, 818458752,...; F^4: [1, 4, 14, (60), 401, 5208, 146520, 8829536, 1100282640,...; F^5: [1, 5, 20, 90, (585), 7121, 190770, 11236080, 1386816800,...; F^6: [1, 6, 27, 128, 819, (9390), 238949, 13733004, 1678197564,...; F^7: [1, 7, 35, 175, 1113, 12089, (291641), 16326885, 1974570178,...; F^8: [1, 8, 44, 232, 1478, 15304, 349532, (19025176), 2276089889,...; F^9: [1, 9, 54, 300, 1926, 19134, 413424, 21836340, (2582923185),...; dividing each diagonal term in row n by (n+1) gives a(n) for n>=0. The diagonal above the main diagonal gives coefficients of l.g.f.: log(A(x)) = x + 5*x^2/2 + 37*x^3/3 + 401*x^4/4 + 7121*x^5/5 +... MATHEMATICA max = 15; s = x/Sum[2^(k*(k-1)/2)*x^k, {k, 0, max}] + O[x]^(max+2); (1/x)*InverseSeries[s] + O[x]^(max+1) // CoefficientList[#, x]& (* Jean-François Alcover, Sep 03 2017 *) PROG (PARI) a(n)=polcoeff(1/x*serreverse(x/sum(k=0, n, 2^(k*(k-1)/2)*x^k +x*O(x^n))), n) CROSSREFS Cf. A136653. Sequence in context: A080290 A365777 A259843 * A364621 A145161 A121422 Adjacent sequences: A136651 A136652 A136653 * A136655 A136656 A136657 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 15 2008 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 December 3 03:10 EST 2023. Contains 367531 sequences. (Running on oeis4.)