A383377 - OEIS

%I #13 May 18 2025 03:20:04

%S 1,1,2,4,6,6,20,46,92,138,276,676,1476,3332,5670,11574,27262,61952,

%T 135354,222848,549226,1319282,3068894,6449978,10987080,27779594,

%U 67311236,157054012,313271538,579149708,1452091208,3548249288,7866783754,16098393372,32442930610,78084645030,180671169756

%N 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).

%C Compare to C(x) = Sum_{n>=0} x^n * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

%C Conjecture: a(n) is even for n > 1.

%H Paul D. Hanna, <a href="/A383377/b383377.txt">Table of n, a(n) for n = 0..515</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) A(x) = Sum_{n>=0} x^n * abs( 1/A(x)^n ).

%F (2) a(n) = Sum_{k=0..n} abs( [x^k] 1/A(x)^(n-k) ) for n >= 0.

%e 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 + ...

%e The coefficients in 1/A(x)^n begin

%e   n = 1: [1, -1, -1, -1,  1,   5,  -11,  -17, ...];

%e   n = 2: [1, -2, -1,  0,  5,  10,  -33,  -24, ...];

%e   n = 3: [1, -3,  0,  2,  9,   9,  -70,   -6, ...];

%e   n = 4: [1, -4,  2,  4, 11,   0, -116,   64, ...];

%e   n = 5: [1, -5,  5,  5, 10, -16, -160,  210, ...];

%e   n = 6: [1, -6,  9,  4,  6, -36, -190,  444, ...];

%e   n = 7: [1, -7, 14,  0,  0, -56, -196,  762, ...];

%e   n = 8: [1, -8, 20, -8, -6, -72, -172, 1144, ...];

%e   ...

%e The table of unsigned coefficients that form the series abs(1/A(x)^n) begins

%e   n = 0: [1,  0,  0,  0,  0,  0,   0,    0,    0,    0, ...];

%e   n = 1: [1,  1,  1,  1,  1,  5,  11,   17,    7,   69, ...];

%e   n = 2: [1,  2,  1,  0,  5, 10,  33,   24,   33,  218, ...];

%e   n = 3: [1,  3,  0,  2,  9,  9,  70,    6,  123,  377, ...];

%e   n = 4: [1,  4,  2,  4, 11,  0, 116,   64,  253,  452, ...];

%e   n = 5: [1,  5,  5,  5, 10, 16, 160,  210,  375,  325, ...];

%e   n = 6: [1,  6,  9,  4,  6, 36, 190,  444,  399,  102, ...];

%e   n = 7: [1,  7, 14,  0,  0, 56, 196,  762,  203,  847, ...];

%e   n = 8: [1,  8, 20,  8,  6, 72, 172, 1144,  349, 1792, ...];

%e   n = 9: [1,  9, 27, 21,  9, 81, 117, 1557, 1386, 2644, ...];

%e   n =10: [1, 10, 35, 40,  5, 82,  35, 1960, 3010, 2920, ...];

%e   ...

%e in which the antidiagonal sums equal this sequence

%e   a(0) = 1 = 1;

%e   a(1) = 0 + 1 = 1;

%e   a(2) = 0 + 1 + 1 = 2;

%e   a(3) = 0 + 1 + 2 + 1 = 4;

%e   a(4) = 0 + 1 + 1 + 3 + 1 = 6;

%e   a(5) = 0 + 1 + 0 + 0 + 4 + 1 = 6;

%e   a(6) = 0 + 5 + 5 + 2 + 2 + 5 + 1 = 20;

%e   a(7) = 0 + 11 + 10 + 9 + 4 + 5 + 6 + 1 = 46;

%e   a(8) = 0 + 17 + 33 + 9 + 11 + 5 + 9 + 7 + 1 = 92;

%e   a(9) = 0 + 7 + 24 + 70 + 0 + 10 + 4 + 14 + 8 + 1 = 138;

%e   a(10) = 0 + 69 + 33 + 6 + 116 + 16 + 6 + 0 + 20 + 9 + 1 = 276;

%e   ...

%e illustrating a(n) = Sum_{k=0..n} abs( [x^(n-k)] 1/A(x)^k ) for n >= 0.

%o (PARI) {a(n) = my(V=[1], A);

%o for(i=1,n, V = concat(V,0); A = Ser(V);

%o V[#V] = -polcoef(truncate(A) - 1 - sum(m=1,#V+1, x^m * Ser(abs(Vec( 1/A^m ))) ),#V-1) );V[n+1]}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A382122.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 15 2025