A383377 G.f. satisfies A(x) = Sum_{n>=0} x^n * abs(1/A(x)^n), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).
1, 1, 2, 4, 6, 6, 20, 46, 92, 138, 276, 676, 1476, 3332, 5670, 11574, 27262, 61952, 135354, 222848, 549226, 1319282, 3068894, 6449978, 10987080, 27779594, 67311236, 157054012, 313271538, 579149708, 1452091208, 3548249288, 7866783754, 16098393372, 32442930610, 78084645030, 180671169756
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 6*x^5 + 20*x^6 + 46*x^7 + 92*x^8 + 138*x^9 + 276*x^10 + 676*x^11 + 1476*x^12 + ...
The coefficients in 1/A(x)^n begin
n = 1: [1, -1, -1, -1, 1, 5, -11, -17, ...];
n = 2: [1, -2, -1, 0, 5, 10, -33, -24, ...];
n = 3: [1, -3, 0, 2, 9, 9, -70, -6, ...];
n = 4: [1, -4, 2, 4, 11, 0, -116, 64, ...];
n = 5: [1, -5, 5, 5, 10, -16, -160, 210, ...];
n = 6: [1, -6, 9, 4, 6, -36, -190, 444, ...];
n = 7: [1, -7, 14, 0, 0, -56, -196, 762, ...];
n = 8: [1, -8, 20, -8, -6, -72, -172, 1144, ...];
...
The table of unsigned coefficients that form the series abs(1/A(x)^n) begins
n = 0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n = 1: [1, 1, 1, 1, 1, 5, 11, 17, 7, 69, ...];
n = 2: [1, 2, 1, 0, 5, 10, 33, 24, 33, 218, ...];
n = 3: [1, 3, 0, 2, 9, 9, 70, 6, 123, 377, ...];
n = 4: [1, 4, 2, 4, 11, 0, 116, 64, 253, 452, ...];
n = 5: [1, 5, 5, 5, 10, 16, 160, 210, 375, 325, ...];
n = 6: [1, 6, 9, 4, 6, 36, 190, 444, 399, 102, ...];
n = 7: [1, 7, 14, 0, 0, 56, 196, 762, 203, 847, ...];
n = 8: [1, 8, 20, 8, 6, 72, 172, 1144, 349, 1792, ...];
n = 9: [1, 9, 27, 21, 9, 81, 117, 1557, 1386, 2644, ...];
n =10: [1, 10, 35, 40, 5, 82, 35, 1960, 3010, 2920, ...];
...
in which the antidiagonal sums equal this sequence
a(0) = 1 = 1;
a(1) = 0 + 1 = 1;
a(2) = 0 + 1 + 1 = 2;
a(3) = 0 + 1 + 2 + 1 = 4;
a(4) = 0 + 1 + 1 + 3 + 1 = 6;
a(5) = 0 + 1 + 0 + 0 + 4 + 1 = 6;
a(6) = 0 + 5 + 5 + 2 + 2 + 5 + 1 = 20;
a(7) = 0 + 11 + 10 + 9 + 4 + 5 + 6 + 1 = 46;
a(8) = 0 + 17 + 33 + 9 + 11 + 5 + 9 + 7 + 1 = 92;
a(9) = 0 + 7 + 24 + 70 + 0 + 10 + 4 + 14 + 8 + 1 = 138;
a(10) = 0 + 69 + 33 + 6 + 116 + 16 + 6 + 0 + 20 + 9 + 1 = 276;
...
illustrating a(n) = Sum_{k=0..n} abs( [x^(n-k)] 1/A(x)^k ) for n >= 0.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..515
Crossrefs
Cf. A382122.
Programs
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PARI
{a(n) = my(V=[1], A); for(i=1,n, V = concat(V,0); A = Ser(V); V[#V] = -polcoef(truncate(A) - 1 - sum(m=1,#V+1, x^m * Ser(abs(Vec( 1/A^m ))) ),#V-1) );V[n+1]} for(n=0,40,print1(a(n),", "))
Formula
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} x^n * abs( 1/A(x)^n ).
(2) a(n) = Sum_{k=0..n} abs( [x^k] 1/A(x)^(n-k) ) for n >= 0.
Comments