A158119 -id:A158119 - cp's OEIS frontend

A157308 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)) = g.f. of A155585 have zeros for every other coefficient after initial terms; g.f. of dual sequence A157309 satisfies the same condition.

1, 1, -1, 0, 3, 0, -38, 0, 947, 0, -37394, 0, 2120190, 0, -162980012, 0, 16330173251, 0, -2070201641498, 0, 324240251016266, 0, -61525045423103316, 0, 13913915097436287598, 0, -3698477457114061621492, 0

Offset: 0

Paul D. Hanna, Mar 11 2009

sign

After initial 2 terms, reversing signs yields A157310.

Conjecture: a(m) == 1 (mod 2) iff m is a power of 2 or m=0. [Paul D. Hanna, Mar 17 2009]

			G.f.: A(x) = 1 + x - x^2 + 3*x^4 - 38*x^6 + 947*x^8 - 37394*x^10 +-...
RELATED FUNCTIONS.
If F(x) = A(x*F(x)) then F(x) = o.g.f. of A155585:
A155585 = [1,1,0,-2,0,16,0,-272,0,7936,0,-353792,0,...];
...
If G(x) = A(x*G(x))/(1+x) then G(x) = o.g.f. of A122045:
A122045 = [1,0,-1,0,5,0,-61,0,1385,0,-50521,0,2702765,0,...];
...

Cf. A157309, A157310, A157304, A157305, A155585, A122045 (Euler numbers).

Cf. A158119. [Paul D. Hanna, Mar 17 2009]

terms = 28;
F[x_] = Sum[n! x^n/Product[(1 + 2k x), {k, 1, n}], {n, 0, terms+1}] + O[x]^(terms+1);
A[x_] = x/InverseSeries[x F[x]];
CoefficientList[A[x], x][[1 ;; terms]] (* Jean-François Alcover, Jul 26 2018 *)

{a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==0, A=concat(A, 0);); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); Vec(x/serreverse(x*Ser(A)))[n+1]}

Let F(x) = o.g.f. of A155585, then o.g.f. A(x) satisfies:
A(x) = x/serreverse(x*F(x));
A(x) = 2x + F( -x/(A(x) - 2x) );
A(x) = F(x/A(x));
F(x) = A(x*F(x));
where A155585 is defined by e.g.f. exp(x)/cosh(x).
...
Let G(x) = o.g.f. of A122045, then o.g.f. A(x) satisfies:
A(x) = x + x/serreverse(x*G(x));
A(x) = x + G( x/(A(x) - x) );
G(x) = A(x*G(x))/(1+x);
where A122045 is the Euler numbers.
...
O.g.f.: A(x) = 2*(1+x) - H(x) where H(x) = g.f. of A157310.

A338633 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^3x/(A(x) - 3^3x/(A(x) - 4^3x/(A(x) - 5^3x/(A(x) - 6^3*x/(A(x) - ...)))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 7, 250, 21867, 3725702, 1096355494, 513875333940, 361121449989171, 362961084011245198, 502496711191618404882, 929337000359116522329132, 2238572532534241145084855934, 6875030222633195280825967544508, 26436454884630260855874989243890732

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

Compare to the continued fraction relation for the g.f. of A158119 and A338634.

			G.f.: A(x) = 1 + x + 7*x^2 + 250*x^3 + 21867*x^4 + 3725702*x^5 + 1096355494*x^6 + 513875333940*x^7 + 361121449989171*x^8 + 362961084011245198*x^9 + ...
where
1 = A(x) - x/(A(x) - 2^3*x/(A(x) - 3^3*x/(A(x) - 4^3*x/(A(x) - 5^3*x/(A(x) - 6^3*x/(A(x) - 7^3*x/(A(x) - 8^3*x/(A(x) - 9^3*x/(A(x) - ...))))))))), a continued fraction relation.

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A000699, A158119, A216966, A338634.

{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (#A-i+1)^3*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }
for(n=0,20,print1(a(n),", "))

For n > 0, a(n) is odd iff n is a power of 2 (conjecture).
From Vaclav Kotesovec, Nov 12 2020: (Start)
a(n) ~ sqrt(3/(2*Pi)) * (6*Gamma(2/3)/Gamma(1/3)^2)^(3*n + 3/2) * (n!)^3 / sqrt(n).
a(n) ~ 2^(6*n + 4) * 3^(3*n/2 + 5/4) * Pi^(3*n + 5/2) * n^(3*n + 1) / Gamma(1/3)^(9*(n + 1/2)) / exp(3*n). (End)

A338634 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^4x/(A(x) - 3^4x/(A(x) - 4^4x/(A(x) - 5^4x/(A(x) - 6^4*x/(A(x) - ... )))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 15, 1490, 472475, 367254494, 596838469302, 1812465211795364, 9460229930323620755, 79588323526110945959270, 1025816228173896271039326050, 19441688693651416990291991566332, 523762848713992063145153491388390686, 19495503038639783268900576813041922912172

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

			G.f.: A(x) = 1 + x + 15*x^2 + 1490*x^3 + 472475*x^4 + 367254494*x^5 + 596838469302*x^6 + 1812465211795364*x^7 + 9460229930323620755*x^8 + ...
where
1 = A(x) - x/(A(x) - 2^4*x/(A(x) - 3^4*x/(A(x) - 4^4*x/(A(x) - 5^4*x/(A(x) - 6^4*x/(A(x) - 7^4*x/(A(x) - 8^4*x/(A(x) - 9^4*x/(A(x) - 10^4*x/(A(x) - ... )))))))))), a continued fraction relation.

Paul D. Hanna, Table of n, a(n) for n = 0..130

Cf. A000699, A158119, A227887, A338633, A338635.

{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (#A-i+1)^4*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }
for(n=0,20,print1(a(n),", "))

For n > 0, a(n) is odd iff n is a power of 2 (conjecture).
From Vaclav Kotesovec, Nov 12 2020: (Start)
a(n) ~ sqrt(2/Pi) * (8*sqrt(Pi) / Gamma(1/4)^2)^(4*n + 2) * (n!)^4 / sqrt(n).
a(n) ~ 2^(12*n + 17/2) * Pi^(2*n + 5/2) * n^(4*n + 3/2) / (Gamma(1/4)^(8*n + 4) * exp(4*n)). (End)

A158120 Unsigned bisection of A157304 and A157305.

Original entry on oeis.org

1, 2, 26, 1378, 141202, 22716418, 5218302090, 1619288968386, 653379470919714, 333014944014777730, 209463165121436380282, 159492000935562428176162, 144654795258284936534929586, 154140229756873813307283828098

Offset: 0

Paul D. Hanna, Mar 12 2009

nonn

			G.f.: A(x) = 1 + 2*x + 26*x^2 + 1378*x^3 + 141202*x^4 +...
RELATED FUNCTIONS.
G.f. of A157305, B(x) = x + A(-x^2), satisfies the condition
that both B(x) and F(x) = B(x*F(x)^2) = o.g.f. of A157307
have zeros for every other coefficient after initial terms:
A157305 = [1,1,-2,0,26,0,-1378,0,141202,0,-22716418,0,...];
A157307 = [1,1,0,-7,0,242,0,-17771,0,2189294,0,-404590470,0,...].
...
G.f. of A157304, C(x) = 2+x - A(-x^2), satisfies the condition
that both C(x) and G(x) = C(x/G(x)^2) = o.g.f. of A157302
have zeros for every other coefficient after initial terms:
A157308 = [1,1,2,0,-26,0,1378,0,-141202,0,22716418,0,...];
A157302 = [1,1,0,-5,0,183,0,-14352,0,1857199,0,-355082433,0,...].
...

Paul D. Hanna, Table of n, a(n) for n=0..50

Cf. A157304, A157305, A157302, A157307, A158119.

{a(n)=local(A=[1, 1]); for(i=1, 2*n, if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); (-1)^n*Vec(x/serreverse(x*Ser(A)))[2*n+1]}

A338632 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3x/(A(x) - 5x/(A(x) - 7x/(A(x) - 9x/(A(x) - ...))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 2, 14, 166, 2714, 55866, 1377942, 39493518, 1288115570, 47086272754, 1906554619166, 84711219819062, 4098314765667082, 214489189682087594, 12075596389435432230, 727783484288200558110, 46755528594469120151010, 3189788089674119448202722

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

			G.f. A(x) = 1 + x + 2*x^2 + 14*x^3 + 166*x^4 + 2714*x^5 + 55866*x^6 + 1377942*x^7 + 39493518*x^8 + 1288115570*x^9 + 47086272754*x^10 + ...
where
1 = A(x) - x/(A(x) - 3*x/(A(x) - 5*x/(A(x) - 7*x/(A(x) - 9*x/(A(x) - 11*x/(A(x) - 13*x/(A(x) - 15*x/(A(x) - 17*x/(A(x) - 19*x/(A(x) - ...)))))))))), a continued fraction relation.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A000699, A158119, A338636.

Cf. A191107, A186776.

{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (2*(#A-i)+1)*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }
for(n=0,20,print1(a(n),", "))

a(n) = 2 (mod 4) for n > 1 (conjecture).
For n > 0, a(n) = 1 (mod 3) iff n = A191107(k) for some k >= 1 (conjecture).
For n > 0, a(n) = 2 (mod 3) iff n = A186776(k) for some k >= 2 where A186776 is the Stanley sequence S(0,2) (conjecture).
a(n) ~ 2^(2*n) * n^(n - 1/2) / (sqrt(Pi) * exp(n + 1/2)). - Vaclav Kotesovec, Nov 12 2020

A338635 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3x/(A(x) - 6x/(A(x) - 10x/(A(x) - 15x/(A(x) - 21x/(A(x) - 28x/(A(x) - ... - (n(n+1)/2)x/(A(x) - ...))))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 2, 17, 274, 6749, 231276, 10465440, 604220826, 43388420549, 3797054582794, 398157728106929, 49311011342018168, 7124133759620985652, 1187818792835133749984, 226420783437860189825400, 48936975180367428260159850, 11904986360488865549641429797, 3238569202146221391019821488694

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 17*x^3 + 274*x^4 + 6749*x^5 + 231276*x^6 + 10465440*x^7 + 604220826*x^8 + 43388420549*x^9 + 3797054582794*x^10 + ...
where
1 = A(x) - x/(A(x) - 3*x/(A(x) - 6*x/(A(x) - 10*x/(A(x) - 15*x/(A(x) - 21*x/(A(x) - 28*x/(A(x) - 36*x/(A(x) - 45*x/(A(x) - 55*x/(A(x) - ...)))))))))), a continued fraction relation in which the triangular numbers appear as coefficients.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A000699, A158119, A338633, A338634, A118113.

{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (#A-i+1)*(#A-i+2)/2*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }
for(n=0,20,print1(a(n),", "))

For n>0, a(n) is odd iff n = A118113(k) for some k >= 1, where A118113(k) = 2*Fibbinary(k) + 1 (conjecture).

a(n) ~ 2^(3*n + 5) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * exp(2*n)). - Vaclav Kotesovec, Nov 12 2020

A338636 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2x/(A(x) - 5^2x/(A(x) - 7^2x/(A(x) - 9^2x/(A(x) - ...))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 8, 272, 19480, 2353568, 429016872, 110046546096, 37825128764472, 16793443888112960, 9358539226503013960, 6397425528561882140240, 5264539843826571207135320, 5134140710880677886077086432, 5855644914993764696284947092840

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

			G.f. A(x) = 1 + x + 8*x^2 + 272*x^3 + 19480*x^4 + 2353568*x^5 + 429016872*x^6 + 110046546096*x^7 + 37825128764472*x^8 + 16793443888112960*x^9 + ...
where
1 = A(x) - x/(A(x) - 3^2*x/(A(x) - 5^2*x/(A(x) - 7^2*x/(A(x) - 9^2*x/(A(x) - 11^2*x/(A(x) - 13^2*x/(A(x) - 15^2*x/(A(x) - 17^2*x/(A(x) - 19^2*x/(A(x) - ...)))))))))), a continued fraction relation.

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A338632, A158119.

Cf. A191107, A186776.

{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (2*(#A-i)+1)^2*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }
for(n=0,20,print1(a(n),", "))

a(n) = 0 (mod 8) for n > 1 (conjecture).
For n > 0, a(n) = 1 (mod 3) iff n = A191107(k) for some k >= 1 (conjecture).
For n > 0, a(n) = 2 (mod 3) iff n = A186776(k) for some k >= 2 where A186776 is the Stanley sequence S(0,2) (conjecture).
a(n) ~ 2^(6*n + 1) * n^(2*n - 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Nov 12 2020

A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(xG(x)) = Sum_{n>=0} A003701(n)x^n.

Original entry on oeis.org

1, 1, 1, 0, 3, 0, 38, 0, 947, 0, 37394, 0, 2120190, 0, 162980012, 0, 16330173251, 0, 2070201641498, 0, 324240251016266, 0, 61525045423103316, 0, 13913915097436287598, 0, 3698477457114061621492, 0

Offset: 0

Paul D. Hanna, Feb 07 2010

nonn

The e.g.f. of A003701 is exp(x)/cos(x) = Sum_{n>=0} A003701(n)*x^n/n!.

Compare to A157308 and A157310.

			G.f.: A(x) = 1 + x + x^2 + 3*x^4 + 38*x^6 + 947*x^8 + 37394*x^10 +...
where G(x) = A(x*G(x)) is the o.g.f. of A003701:
G(x) = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 +...
while the e.g.f. of A003701 is given by:
exp(x)/cos(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 12*x^4/4! + 36*x^5/5! +...

Cf. A003701, A158119, A157308, A157310.

{a(n)=local(X=x+x*O(x^n),G=sum(m=0,n,m!*polcoeff(exp(X)/cos(X),m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}

a(n) = |A157308(n)| = |A157310(n)| for n>=0.
a(2n) = A158119(n) for n>=0; a(2n-1) = 0 for n>=2, with a(1)=1.
G.f. A = A(x) satisfies: A(x) = 1/(1-x/A - (x/A)^2/(1-x/A - 2^2*(x/A)^2/(1-x/A - 3^2*(x/A)^2/(1-x/A - 4^2*(x/A)^2/(1-x/A - 5^2*(x/A)^2/(1-x/A -...)))))), a recursive continued fraction. [From Paul D. Hanna, Jan 05 2012]

cp's OEIS Frontend

A157308 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)) = g.f. of A155585 have zeros for every other coefficient after initial terms; g.f. of dual sequence A157309 satisfies the same condition.

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A338633 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^3*x/(A(x) - 3^3*x/(A(x) - 4^3*x/(A(x) - 5^3*x/(A(x) - 6^3*x/(A(x) - ...)))))), a continued fraction relation.

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A338634 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^4*x/(A(x) - 3^4*x/(A(x) - 4^4*x/(A(x) - 5^4*x/(A(x) - 6^4*x/(A(x) - ... )))))), a continued fraction relation.

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A158120 Unsigned bisection of A157304 and A157305.

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A338632 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3*x/(A(x) - 5*x/(A(x) - 7*x/(A(x) - 9*x/(A(x) - ...))))), a continued fraction relation.

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A338635 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3*x/(A(x) - 6*x/(A(x) - 10*x/(A(x) - 15*x/(A(x) - 21*x/(A(x) - 28*x/(A(x) - ... - (n*(n+1)/2)*x/(A(x) - ...))))))), a continued fraction relation.

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A338636 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2*x/(A(x) - 5^2*x/(A(x) - 7^2*x/(A(x) - 9^2*x/(A(x) - ...))))), a continued fraction relation.

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A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A003701(n)*x^n.

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A338633 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^3x/(A(x) - 3^3x/(A(x) - 4^3x/(A(x) - 5^3x/(A(x) - 6^3*x/(A(x) - ...)))))), a continued fraction relation.

A338634 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^4x/(A(x) - 3^4x/(A(x) - 4^4x/(A(x) - 5^4x/(A(x) - 6^4*x/(A(x) - ... )))))), a continued fraction relation.

A338632 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3x/(A(x) - 5x/(A(x) - 7x/(A(x) - 9x/(A(x) - ...))))), a continued fraction relation.

A338635 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3x/(A(x) - 6x/(A(x) - 10x/(A(x) - 15x/(A(x) - 21x/(A(x) - 28x/(A(x) - ... - (n(n+1)/2)x/(A(x) - ...))))))), a continued fraction relation.

A338636 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2x/(A(x) - 5^2x/(A(x) - 7^2x/(A(x) - 9^2x/(A(x) - ...))))), a continued fraction relation.

A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(xG(x)) = Sum_{n>=0} A003701(n)x^n.