This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383377 #13 May 18 2025 03:20:04
%S A383377 1,1,2,4,6,6,20,46,92,138,276,676,1476,3332,5670,11574,27262,61952,
%T A383377 135354,222848,549226,1319282,3068894,6449978,10987080,27779594,
%U A383377 67311236,157054012,313271538,579149708,1452091208,3548249288,7866783754,16098393372,32442930610,78084645030,180671169756
%N 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).
%C A383377 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 A383377 Conjecture: a(n) is even for n > 1.
%H A383377 Paul D. Hanna, <a href="/A383377/b383377.txt">Table of n, a(n) for n = 0..515</a>
%F A383377 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A383377 (1) A(x) = Sum_{n>=0} x^n * abs( 1/A(x)^n ).
%F A383377 (2) a(n) = Sum_{k=0..n} abs( [x^k] 1/A(x)^(n-k) ) for n >= 0.
%e A383377 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 A383377 The coefficients in 1/A(x)^n begin
%e A383377 n = 1: [1, -1, -1, -1, 1, 5, -11, -17, ...];
%e A383377 n = 2: [1, -2, -1, 0, 5, 10, -33, -24, ...];
%e A383377 n = 3: [1, -3, 0, 2, 9, 9, -70, -6, ...];
%e A383377 n = 4: [1, -4, 2, 4, 11, 0, -116, 64, ...];
%e A383377 n = 5: [1, -5, 5, 5, 10, -16, -160, 210, ...];
%e A383377 n = 6: [1, -6, 9, 4, 6, -36, -190, 444, ...];
%e A383377 n = 7: [1, -7, 14, 0, 0, -56, -196, 762, ...];
%e A383377 n = 8: [1, -8, 20, -8, -6, -72, -172, 1144, ...];
%e A383377 ...
%e A383377 The table of unsigned coefficients that form the series abs(1/A(x)^n) begins
%e A383377 n = 0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e A383377 n = 1: [1, 1, 1, 1, 1, 5, 11, 17, 7, 69, ...];
%e A383377 n = 2: [1, 2, 1, 0, 5, 10, 33, 24, 33, 218, ...];
%e A383377 n = 3: [1, 3, 0, 2, 9, 9, 70, 6, 123, 377, ...];
%e A383377 n = 4: [1, 4, 2, 4, 11, 0, 116, 64, 253, 452, ...];
%e A383377 n = 5: [1, 5, 5, 5, 10, 16, 160, 210, 375, 325, ...];
%e A383377 n = 6: [1, 6, 9, 4, 6, 36, 190, 444, 399, 102, ...];
%e A383377 n = 7: [1, 7, 14, 0, 0, 56, 196, 762, 203, 847, ...];
%e A383377 n = 8: [1, 8, 20, 8, 6, 72, 172, 1144, 349, 1792, ...];
%e A383377 n = 9: [1, 9, 27, 21, 9, 81, 117, 1557, 1386, 2644, ...];
%e A383377 n =10: [1, 10, 35, 40, 5, 82, 35, 1960, 3010, 2920, ...];
%e A383377 ...
%e A383377 in which the antidiagonal sums equal this sequence
%e A383377 a(0) = 1 = 1;
%e A383377 a(1) = 0 + 1 = 1;
%e A383377 a(2) = 0 + 1 + 1 = 2;
%e A383377 a(3) = 0 + 1 + 2 + 1 = 4;
%e A383377 a(4) = 0 + 1 + 1 + 3 + 1 = 6;
%e A383377 a(5) = 0 + 1 + 0 + 0 + 4 + 1 = 6;
%e A383377 a(6) = 0 + 5 + 5 + 2 + 2 + 5 + 1 = 20;
%e A383377 a(7) = 0 + 11 + 10 + 9 + 4 + 5 + 6 + 1 = 46;
%e A383377 a(8) = 0 + 17 + 33 + 9 + 11 + 5 + 9 + 7 + 1 = 92;
%e A383377 a(9) = 0 + 7 + 24 + 70 + 0 + 10 + 4 + 14 + 8 + 1 = 138;
%e A383377 a(10) = 0 + 69 + 33 + 6 + 116 + 16 + 6 + 0 + 20 + 9 + 1 = 276;
%e A383377 ...
%e A383377 illustrating a(n) = Sum_{k=0..n} abs( [x^(n-k)] 1/A(x)^k ) for n >= 0.
%o A383377 (PARI) {a(n) = my(V=[1], A);
%o A383377 for(i=1,n, V = concat(V,0); A = Ser(V);
%o A383377 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 A383377 for(n=0,40,print1(a(n),", "))
%Y A383377 Cf. A382122.
%K A383377 nonn
%O A383377 0,3
%A A383377 _Paul D. Hanna_, May 15 2025