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 A382122 #13 Mar 28 2025 04:38:23
%S A382122 1,1,3,12,49,202,838,3486,14575,60820,254406,1061438,4444802,18602018,
%T A382122 78066384,326985608,1365996909,5697914836,23752394338,99027785702,
%U A382122 413203462516,1726164299990,7219911692522,30228722494504,126658682953328,530772842793396,2224199143900798,9319843329508200,39051457052597480
%N A382122 G.f. satisfies Sum_{n>=0} x^n * abs(1/A(x)^n) = C(x), where C(x) = 1 + x*C(x)^2 and abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).
%C A382122 Compare to Sum_{n>=0} x^n * C(x)^n = C(x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%C A382122 Conjecture: for n > 0, a(n) is odd iff n = 2^k for k >= 0.
%H A382122 Paul D. Hanna, <a href="/A382122/b382122.txt">Table of n, a(n) for n = 0..521</a>
%F A382122 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A382122 (1) Sum_{n>=0} x^n * abs(1/A(x)^n) = C(x), where C(x) = 1 + x*C(x)^2.
%F A382122 (2) Sum_{k=0..n} abs( [x^k] 1/A(x)^(n-k) ) = binomial(2*n+1,n)/(2*n+1) for n >= 0.
%F A382122 a(n) ~ c * d^n, where d = 4.1935797816358..., c = 0.142779... - _Vaclav Kotesovec_, Mar 28 2025
%e A382122 G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 49*x^4 + 202*x^5 + 838*x^6 + 3486*x^7 + 14575*x^8 + 60820*x^9 + 254406*x^10 + 1061438*x^11 + 4444802*x^12 + ...
%e A382122 Below we illustrate the defining property of this sequence.
%e A382122 The coefficients in 1/A(x)^n begin
%e A382122 1: [1, -1, -2, -7, -24, -84, -298, -1063, ...];
%e A382122 2: [1, -2, -3, -10, -30, -92, -283, -858, ...];
%e A382122 3: [1, -3, -3, -10, -24, -57, -119, -156, ...];
%e A382122 4: [1, -4, -2, -8, -11, -4, 82, 568, ...];
%e A382122 5: [1, -5, 0, -5, 5, 49, 250, 1060, ...];
%e A382122 6: [1, -6, 3, -2, 21, 90, 348, 1224, ...];
%e A382122 7: [1, -7, 7, 0, 35, 112, 364, 1070, ...];
%e A382122 8: [1, -8, 12, 0, 46, 112, 304, 672, ...];
%e A382122 9: [1, -9, 18, -3, 54, 90, 186, 135, ...];
%e A382122 10: [1, -10, 25, -10, 60, 48, 35, -430, ...];
%e A382122 ...
%e A382122 The table of unsigned coefficients that form the series abs(1/A(x)^n) begins
%e A382122 0: [1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e A382122 1: [1, 1, 2, 7, 24, 84, 298, 1063, 3858, ...];
%e A382122 2: [1, 2, 3, 10, 30, 92, 283, 858, 2646, ...];
%e A382122 3: [1, 3, 3, 10, 24, 57, 119, 156, 144, ...];
%e A382122 4: [1, 4, 2, 8, 11, 4, 82, 568, 2578, ...];
%e A382122 5: [1, 5, 0, 5, 5, 49, 250, 1060, 3800, ...];
%e A382122 6: [1, 6, 3, 2, 21, 90, 348, 1224, 3654, ...];
%e A382122 7: [1, 7, 7, 0, 35, 112, 364, 1070, 2394, ...];
%e A382122 8: [1, 8, 12, 0, 46, 112, 304, 672, 469, ...];
%e A382122 9: [1, 9, 18, 3, 54, 90, 186, 135, 1629, ...];
%e A382122 10: [1, 10, 25, 10, 60, 48, 35, 430, 3465, ...];
%e A382122 ...
%e A382122 the antidiagonals of which add to the Catalan numbers (A000108):
%e A382122 1 = 1;
%e A382122 0 + 1 = 1;
%e A382122 0 + 1 + 1 = 2;
%e A382122 0 + 2 + 2 + 1 = 5;
%e A382122 0 + 7 + 3 + 3 + 1 = 14;
%e A382122 0 + 24 + 10 + 3 + 4 + 1 = 42;
%e A382122 0 + 84 + 30 + 10 + 2 + 5 + 1 = 132;
%e A382122 0 + 298 + 92 + 24 + 8 + 0 + 6 + 1 = 429;
%e A382122 0 + 1063 + 283 + 57 + 11 + 5 + 3 + 7 + 1 = 1430;
%e A382122 0 + 3858 + 858 + 119 + 4 + 5 + 2 + 7 + 8 + 1 = 4862;
%e A382122 ...
%o A382122 (PARI) {a(n) = my(V=[1,1], A, C = (1/x)*serreverse(x - x^2 +x^4*O(x^n)));
%o A382122 for(i=1,n, V = concat(V,'t); A = Ser(V);
%o A382122 V[#V] = 't + polcoef(C - sum(m=1,#V+1, x^m * Ser(abs(Vec( 1/A^m ))) ),#V) );V[n+1]}
%o A382122 for(n=0,30,print1(a(n),", "))
%Y A382122 Cf. A000108, A382123.
%K A382122 nonn
%O A382122 0,3
%A A382122 _Paul D. Hanna_, Mar 16 2025