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 A136654 G.f.: A(x) = (1/x)*Series_Reversion( x/Sum_{k=0..n} 2^[k(k-1)/2]*x^k ). 2

%I #7 Sep 03 2017 17:58:35

%S 1,1,3,15,117,1565,41663,2378147,286991465,71261033273,35889915535835,

%T 36421251158141399,74222529448186797341,303194457634544530959125,

%U 2480120130065258782050157847,40601998283406419045206334661611

%N G.f.: A(x) = (1/x)*Series_Reversion( x/Sum_{k=0..n} 2^[k(k-1)/2]*x^k ).

%F a(n) = coefficient of x^n in [Sum_{k=0..n} 2^(k(k-1)/2)*x^k]^(n+1)/(n+1).

%e G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 117*x^4 + 1565*x^5 + 41663*x^6 +...

%e Let F(x) = 1 + x + 2x^2 + 8x^3 + 64x^4 + 1024x^5 +...+ 2^(n(n-1)/2)*x^n +..

%e then A(x) = F(x*A(x)), A(x/F(x)) = F(x).

%e a(n) = coefficient of x^n in F(x)^(n+1)/(n+1),

%e as can be seen by the main diagonal in the array of

%e coefficients in the initial powers of F(x):

%e F^1: [(1), 1, 2, 8, 64, 1024, 32768, 2097152, 268435456,...;

%e F^2: [1, (2), 5, 20, 148, 2208, 67904, 4264960, 541216768,...;

%e F^3: [1, 3, (9), 37, 258, 3588, 105704, 6507552, 818458752,...;

%e F^4: [1, 4, 14, (60), 401, 5208, 146520, 8829536, 1100282640,...;

%e F^5: [1, 5, 20, 90, (585), 7121, 190770, 11236080, 1386816800,...;

%e F^6: [1, 6, 27, 128, 819, (9390), 238949, 13733004, 1678197564,...;

%e F^7: [1, 7, 35, 175, 1113, 12089, (291641), 16326885, 1974570178,...;

%e F^8: [1, 8, 44, 232, 1478, 15304, 349532, (19025176), 2276089889,...;

%e F^9: [1, 9, 54, 300, 1926, 19134, 413424, 21836340, (2582923185),...;

%e dividing each diagonal term in row n by (n+1) gives a(n) for n>=0.

%e The diagonal above the main diagonal gives coefficients of l.g.f.:

%e log(A(x)) = x + 5*x^2/2 + 37*x^3/3 + 401*x^4/4 + 7121*x^5/5 +...

%t 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 *)

%o (PARI) a(n)=polcoeff(1/x*serreverse(x/sum(k=0,n,2^(k*(k-1)/2)*x^k +x*O(x^n))),n)

%Y Cf. A136653.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 15 2008

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Last modified February 23 14:24 EST 2024. Contains 370283 sequences. (Running on oeis4.)