%I #32 Oct 17 2019 11:59:52

%S 1,4,1,15,1,12,40,16,1,77,92,24,101,28,204,373,1,36,667,40,575,689,

%T 826,48,393,1582,1379,1937,590,60,6101,64,1,5227,3129,9515,1826,76,

%U 4390,12404,11341,84,18361,88,5875,46320,7844,96,1553,33133,38886,50883,25741,108,25507,44993,82265,91449,15835,120,150162,124,19376,390653,1,104015,29394,136,242217,249506,507789,144,210831,148,33079,647187,593029,711482,47101,160

%N L.g.f.: Sum_{n=-oo..+oo} (x - x^(2*n-1))^(2*n-1) / (2*n-1).

%C Compare l.g.f. to: Sum_{n=-oo..+oo, n<>0} (x - x^n)^n / n = -log(1-x).

%C Here l.g.f. L(x) = Sum_{n>=1} a(n) * x^(2*n-1) / (2*n-1).

%C a(2^n + 1) = 1 for n >= 1 (conjecture).

%H Paul D. Hanna, <a href="/A293129/b293129.txt">Table of n, a(n) for n = 1..2050</a>

%F L.g.f.: Sum_{n=-oo..+oo} (x + x^(2*n-1))^(2*n-1) / (2*n-1) - note the plus sign.

%F L.g.f.: -log(1-x) - Sum_{n=-oo..+oo, n<>0} (x - x^(2*n))^(2*n) / (2*n).

%F L.g.f.: L(x) = P(x) + Q(x) where

%F P(x) = Sum_{n>=1} (x - x^(2*n-1))^(2*n-1) / (2*n-1),

%F Q(x) = Sum_{n>=1} x^((2*n-1)^2) / ( (2*n-1) * (1 - x^(2*n))^(2*n-1) ).

%e L.g.f.: L(x) = x + 4*x^3/3 + x^5/5 + 15*x^7/7 + x^9/9 + 12*x^11/11 + 40*x^13/13 + 16*x^15/15 + x^17/17 + 77*x^19/19 + 92*x^21/21 + 24*x^23/23 + 101*x^25/25 + 28*x^27/27 + 204*x^29/29 + 373*x^31/31 + x^33/33 + 36*x^35/35 + 667*x^37/37 + 40*x^39/39 + 575*x^41/41 + 689*x^43/43 + 826*x^45/45 + 48*x^47/47 + 393*x^49/49 + 1582*x^51/51 + 1379*x^53/53 + 1937*x^55/55 + 590*x^57/57 + 60*x^59/59 +...

%e such that L(x) = Sum_{n=-oo..+oo} (x - x^(2*n-1))^(2*n-1) / (2*n-1).

%e The coefficient of x^(2^n+1)/(2^n+1) in L(x) for n>=1 begins:

%e [4, 1, 1, 1, 1, 1, 1, 1, 1, ...],

%e and it appears that a(k) = 1 only at k = 1 and k = 2^n + 1 (n>=1).

%e We may write L(x) = P(x) + Q(x) where

%e P(x) = (x - x) + (x - x^3)^3/3 + (x - x^5)^5/5 + (x - x^7)^7/7 + (x - x^9)^9/9 + (x - x^11)^11/11 + (x - x^13)^13/13 + (x - x^15)^15/15 + (x - x^17)^17/17 + (x - x^19)^19/19 + (x - x^21)^21/21 +...+ (x - x^(2*n-1))^(2*n-1)/(2*n-1) +...

%e Q(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...

%e Explicitly,

%e P(x) = x^3/3 - 4*x^5/5 + 8*x^7/7 - 11*x^9/9 + x^11/11 + 14*x^13/13 + x^15/15 - 50*x^17/17 + 58*x^19/19 + x^21/21 + x^23/23 - 54*x^25/25 + x^27/27 - 28*x^29/29 + 311*x^31/31 - 340*x^33/33 + x^35/35 + 75*x^37/37 + x^39/39 - 81*x^41/41 + 345*x^43/43 - 44*x^45/45 + x^47/47 - 1427*x^49/49 + 1531*x^51/51 - 52*x^53/53 + 496*x^55/55 - 1253*x^57/57 + x^59/59 + 1343*x^61/61 + x^63/63 - 2924*x^65/65 +...

%e Q(x) = x + 3*x^3/3 + 5*x^5/5 + 7*x^7/7 + 12*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + 51*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + 341*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + 2925*x^65/65 +...

%e The coefficient of x^(2^n+1)/(2^n+1) in P(x) for n>=1 begins:

%e [1, -4, -11, -50, -340, -2924, -169032, -33445208, -21619038032, 1 - A293599(n), ...].

%e The coefficient of x^(2^n+1)/(2^n+1) in Q(x) for n>=1 begins:

%e [3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, ..., A293599(n), ...].

%o (PARI) {a(n) = my(P,Q,Ox = O(x^(2*n+1)));

%o P = sum(m=1,n+1, (x - x^(2*m-1) +Ox)^(2*m-1) / (2*m-1) );

%o Q = sum(m=1,sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );

%o (2*n-1)*polcoeff(P + Q, 2*n-1)}

%o for(n=1,80,print1(a(n),", "))

%Y Cf. A293597 (P(x)), A293598 (Q(x)), A293599, A291937.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 11 2017