A316363 -id:A316363 - cp's OEIS frontend

A317800 G.f. A(x) satisfies: Sum_{n>=1} ( A(x) - (-1)^n * A(-x) )^n * (-1)^n / n = 0.

1, 1, 1, 4, 10, 33, 105, 354, 1214, 4206, 14846, 52750, 189516, 686745, 2506913, 9211226, 34036230, 126426446, 471769950, 1767460752, 6645539212, 25076120890, 94937019050, 360268374124, 1369645176012, 5226326126048, 20039843858208, 76654036799842, 290534140464144, 1123489897863753, 4582416833711249, 17212665701732282, 45565498032190230

Offset: 1

Paul D. Hanna, Aug 09 2018

sign

Odd terms occur at a(2^k - 1) and a(2^k - 2) for k > 1 and at a(1), while a(n) is even elsewhere (conjecture).

First negative term is a(37).

			G.f. A(x) = x + x^2 + x^3 + 4*x^4 + 10*x^5 + 33*x^6 + 105*x^7 + 354*x^8 + 1214*x^9 + 4206*x^10 + ...
Let the series bisections of g.f. A(x) be denoted by
C = (A(x) + A(-x))/2 = x^2 + 4*x^4 + 33*x^6 + 354*x^8 + 4206*x^10 + ...
S = (A(x) - A(-x))/2 = x + x^3 + 10*x^5 + 105*x^7 + 1214*x^9 + 14846*x^11 + ...
then from the definition we have
0 = (2*C) - (2*S)^2/2 + (2*C)^3/3 - (2*S)^4/4 + (2*C)^5/5 - (2*S)^6/6 + (2*C)^7/7 - (2*S)^8/8 + ...
thus  arctanh(2*C) + log(1 - 4*S^2)/2 = 0,
so that  (1 - 2*C)/(1 + 2*C) = 1 - 4*S^2.
RELATED SERIES.
A(A(x)) = x + 2*x^2 + 4*x^3 + 14*x^4 + 52*x^5 + 204*x^6 + 840*x^7 + 3574*x^8 + 15588*x^9 + 69332*x^10 + ... + A316363(n)*x^n + ...
where A(A(x)) = -x + 2 * Series_Reversion( x - x^2/(1 - 2*x^2) ).

Paul D. Hanna, Table of n, a(n) for n = 1..520

Cf. A316363.

/* From: A(x) = S + S^2/(1 - 2*S^2) and A(x) = Series_Reversion(-A(-x)) */
{a(n) = my(A=[1,1],S); for(i=1,n, S=(x*Ser(A) - subst(x*Ser(A),x,-x))/2; A=concat(Vec( S + S^2/(1 - 2*S^2) ),0); if(#A%2==1,A = (A + Vec( serreverse(subst(-x*Ser(A),x,-x)) ) )/2 ); );A[n]}
for(n=1,30, print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2a) Sum_{n>=1} ( A(x) - (-1)^n * A(-x) )^n * (-1)^n / n = 0.
(2b) A(A(x)) = B(x) such that Sum_{n>=1} ( x + (-1)^n * B(x) )^n / n = 0, where B(x) is the o.g.f. of A316363.
(3a) A(A(x)) = -x + 2 * Series_Reversion( x - x^2/(1 - 2*x^2) ).
(3b) A(A(x)) = x + 2 * Series_Reversion( x/sqrt(1 + 2*x^2) - x^2 )^2.
Let C = (A(x) + A(-x))/2 and S = (A(x) - A(-x))/2, then
(4a) arctanh(2*C) + log(1 - 4*S^2)/2 = 0,
(4b) 1 - 4*S^2 = (1 - 2*C)/(1 + 2*C),
(5a) S^2 = C/(1 + 2*C),
(5b) C = S^2/(1 - 2*S^2),
(6a) A(x) = S + S^2/(1 - 2*S^2),
(6b) A(x) = C + sqrt(C/(1 + 2*C)).
(7) 0 = (2*y + y^2 - y^3) - (2 - 2*y + y^2)*A(x) + (1 + y)*A(x)^2 + A(x)^3, where y = -A(-x) = Series_Reversion(A(x)).

A326564 O.g.f. A(x) satisfies: 0 = Sum_{n>=1} (b(n) - A(x))^n * (2*x)^n / n, where b(n) = 1 if n is odd or b(n) = 2 if n is even.

Original entry on oeis.org

1, 1, -2, 7, -26, 102, -420, 1787, -7794, 34666, -156636, 716982, -3317700, 15494156, -72935624, 345701843, -1648489762, 7902956738, -38067806892, 184152092450, -894259126540, 4357738501844, -21302682030328, 104439435098718, -513390992000340, 2529846489669412, -12494572784556440, 61838364112438732, -306647601790749384, 1523380558254732312, -7580755340625743760, 37783723921640161923

Offset: 0

Paul D. Hanna, Aug 28 2019

sign

a(n) is odd iff n = 2^k - 1 for k >= 0.

Signed version of A307413.

			O.g.f.: A(x) = 1 + x - 2*x^2 + 7*x^3 - 26*x^4 + 102*x^5 - 420*x^6 + 1787*x^7 - 7794*x^8 + 34666*x^9 - 156636*x^10 + 716982*x^11 - 3317700*x^12 + 15494156*x^13 - 72935624*x^14 + 345701843*x^15 - 1648489762*x^16 + ...
such that
0 = (1 - A(x))*(2*x) + (2 - A(x))^2*(2*x)^2/2 + (1 - A(x))^3*(2*x)^3/3 + (2 - A(x))^4*(2*x)^4/4 + (1 - A(x))^5*(2*x)^5/5 + (2 - A(x))^6*(2*x)^6/6 + (1 - A(x))^7*(2*x)^7/7 + (2 - A(x))^8*(2*x)^8/8 + (1 - A(x))^9*(2*x)^9/9 + ...
SPECIAL ARGUMENTS.
A( (3 - sqrt(17))/6 ) = 1/2.
A( (15 - sqrt(513))/40 ) = 1/3.
ODD TERMS.
The odd numbers occur at positions 2^n-1 and begin
[1, 1, 7, 1787, 345701843, 37783723921640161923, 1297226675901009799785880946943488094880739, 4359630365907394639251834255689265800511483817161978056491648421720696612963282942355107, ...].

Paul D. Hanna, Table of n, a(n) for n = 0..520

Cf. A316363, A307413.

/* By definition */
{a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, ( ((m+1)%2) + 1 - Ser(A) )^m * (2*x)^m/m), #A)/2); A[n+1]}
for(n=0, 32, print1(a(n), ", "))

/* From: A(x) = 2 - (1/x) * Series_Reversion( x + x^2/(1 - 2*x^2) ) */
{a(n) = my(A = 2 - (1/x)*serreverse(x + x^2/(1 - 2*x^2 +x*O(x^n)))); polcoeff(A,n)}
for(n=0, 32, print1(a(n), ", "))

O.g.f. A = A(x) satisfies:
(1) 0 = Sum_{n>=1} (3 + (-1)^n - 2*A(x))^n * x^n / n.
(2) 0 = arctanh(2*x - 2*x*A) - log(1 - 4*x^2*(2 - A)^2)/2.
(3) 1 - 4*x^2*(2 - A)^2 = (1 + 2*x - 2*x*A) / (1 - 2*x + 2*x*A).
(4) A(x) = 1 + (A - 2)^2*x + 2*(A - 1)*(A - 2)^2*x^2.
(5) 0 = 2*(A - 1)*(A - 2)^2*x^2 + (A - 2)^2*x - (A - 1).
(6) x = ( sqrt( (A-2)^4 + 8*(A-1)^2*(A-2)^2 ) - (A-2)^2 ) / (4*(A-1)*(A-2)^2).
(7) A(x) = 2 - (1/x) * Series_Reversion( x + x^2/(1 - 2*x^2) ).

A353050 G.f. A(x) satisfies: 0 = Sum_{n>=1} (Lucas(n) - A(x))^n * x^n/n, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 2, 5, 298, 18949, 3962150, 1916564344, 2501114025582, 8336852053702202, 73027618049882652700, 1666798946804859125492899, 99738726494828465657124210156, 15634873312495144092899303952245929, 6430416165010536428917103922818349814504

Offset: 0

Paul D. Hanna, Apr 26 2022

nonn

It is conjectured that this is an integer sequence.

This sequence is a special case of a more general conjecture: 0 = Sum_{n>=1} (L(n) - F(x))^n * x^n/n is satisfied by an integer series F(x) if exp( Sum_{n>=1} L(n) * x^n/n ) yields an integer series.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 298*x^3 + 18949*x^4 + 3962150*x^5 + 1916564344*x^6 + 2501114025582*x^7 + 8336852053702202*x^8 + ...
where
0 = (1 - A(x))*x + (3 - A(x))^2*x^2/2 + (4 - A(x))^3*x^3/3 + (7 - A(x))^4*x^4/4 + (11 - A(x))^5*x^5/5 + (18 - A(x))^6*x^6/6 + (29 - A(x))^7*x^7/7 + (47 - A(x))^8*x^8/8 + (76 - A(x))^9*x^9/9 + ... + (Lucas(n) - A(x))^n*x^n/n + ...
Related series.
exp( Sum_{n>=1} A(x)^n * x^n/n ) = 1/(1 - x*A(x)) = 1 + x + 3*x^2 + 10*x^3 + 319*x^4 + 19601*x^5 + 4002282*x^6 + 1924629400*x^7 + 2504975492897*x^8 + 8341867813691252*x^9 + ...
exp( Sum_{n>=1} Lucas(n)^n * x^n/n ) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 + 2470113915*x^7 + 2978904483553*x^8 + 9401949327631932*x^9 + ... + A156216(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A156216, A316363, A000204.

{Lucas(n) = fibonacci(n-1) + fibonacci(n+1)}
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=1,#A, (Lucas(m) - Ser(A))^m*x^m/m), #A));A[n+1]}
for(n=0,20,print1(a(n),", "))

cp's OEIS Frontend

A317800 G.f. A(x) satisfies: Sum_{n>=1} ( A(x) - (-1)^n * A(-x) )^n * (-1)^n / n = 0.

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A326564 O.g.f. A(x) satisfies: 0 = Sum_{n>=1} (b(n) - A(x))^n * (2*x)^n / n, where b(n) = 1 if n is odd or b(n) = 2 if n is even.

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A353050 G.f. A(x) satisfies: 0 = Sum_{n>=1} (Lucas(n) - A(x))^n * x^n/n, where Lucas(n) = A000204(n).

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