cp's OEIS Frontend

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.

Showing 1-10 of 10 results.

A317350 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x) )^n / (2 - (1+x)^n*A(x))^(n+1).

Original entry on oeis.org

1, 1, 2, 12, 200, 4160, 99862, 2767792, 87200166, 3076185774, 120118928740, 5144915483804, 239932734849080, 12106729328331780, 657428964058944716, 38239094075667233528, 2372421500769940561658, 156417910715313378830238, 10923007991339600108590688, 805475337677577620666606928, 62550798567594006106067173708
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2018

Keywords

Comments

G.f. A(x) = G(log(1+x)), where G(x) is the e.g.f. of A317355.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 200*x^4 + 4160*x^5 + 99862*x^6 + 2767792*x^7 + 87200166*x^8 + 3076185774*x^9 + 120118928740*x^10 + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A)  +  ((1+x) - A)/(2 - (1+x)*A)^2  +  ((1+x)^2 - A)^2/(2 - (1+x)^2*A)^3  +  ((1+x)^3 - A)^3/(2 - (1+x)^3*A)^4  +  ((1+x)^4 - A)^4/(2 - (1+x)^4*A)^5  +  ((1+x)^5 - A)^5/(2 - (1+x)^5*A)^6 + ...
Also,
A(x) = 1/(2 + A)  +  ((1+x) + A)/(2 + (1+x)*A)^2  +  ((1+x)^2 + A)^2/(2 + (1+x)^2*A)^3  +  ((1+x)^3 + A)^3/(2 + (1+x)^3*A)^4  +  ((1+x)^4 + A)^4/(2 + (1+x)^4*A)^5  +  ((1+x)^5 + A)^5/(2 + (1+x)^5*A)^6 + ...

Links

Crossrefs

Cf. A317351, A317355.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A = Vec( sum(m=0,#A, ( (1+x)^m - Ser(A) )^m  / (2 - (1+x)^m*Ser(A))^(m+1) ) ) ); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( (1+x)^n - A(x) )^n / (2 - (1+x)^n*A(x))^(n+1),
(2) A(x) = Sum_{n>=0} ( (1+x)^n + A(x) )^n / (2 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317904 = 3.956184203026... and c = 0.14581304299... - Vaclav Kotesovec, Aug 07 2018

A302598 G.f.: Sum_{n>=0} (1 + (1+x)^n)^n / (2 + (1+x)^n)^(n+1).

Original entry on oeis.org

1, 1, 6, 58, 798, 14150, 307076, 7881756, 233536532, 7844786248, 294582696686, 12228351266210, 556017625969246, 27482790417322218, 1467194712330407238, 84134395928742550138, 5157545958316518485420, 336574587493456290969620, 23296320082405927961459550, 1704662916739625989249415610, 131480805016085834305348796128
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2018

Keywords

Comments

The following identity holds for |y| <= 1 and fixed real k > 0:
Sum_{n>=0} (k + y^n)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 58*x^3 + 798*x^4 + 14150*x^5 + 307076*x^6 + 7881756*x^7 + 233536532*x^8 + 7844786248*x^9 + 294582696686*x^10 + ...
such that
A(x) = 1/3  +  (1 + (1+x))/(2 + (1+x))^2  +  (1 + (1+x)^2)^2/(2 + (1+x)^2)^3  +  (1 + (1+x)^3)^3/(2 + (1+x)^3)^4  +  (1 + (1+x)^4)^4/(2 + (1+x)^4)^5  +  (1 + (1+x)^5)^5/(2 + (1+x)^5)^6  +  (1 + (1+x)^6)^6/(2 + (1+x)^6)^7  + ...
Also,
A(x) = 1  +  ((1+x) - 1)/(2 - (1+x))^2  +  ((1+x)^2 - 1)^2/(2 - (1+x)^2)^3  +  ((1+x)^3 - 1)^3/(2 - (1+x)^3)^4  +  ((1+x)^4 - 1)^4/(2 - (1+x)^4)^5  +  ((1+x)^5 - 1)^5/(2 - (1+x)^5)^6  +  ((1+x)^6 - 1)^6/(2 - (1+x)^6)^7 + ...
RELATED INFINITE SERIES.
(1) At x = -1/2: the following sums are equal
S1 = Sum_{n>=0} 2^n * (2^n + 1)^n/(2^(n+1) + 1)^(n+1)
S1 = Sum_{n>=0} (-2)^n * (2^n - 1)^n/(2^(n+1) - 1)^(n+1).
Explicitly,
S1 = 1/3 + 2*3/5^2 + 4*5^2/9^3 + 8*9^3/17^4 + 16*17^4/33^5 + 32*33^5/65^6 + 64*65^6/129^7 + 128*129^7/257^8 + 256*257^8/513^9 + 512*513^9/1025^10 + ...
S1 = 1 - 2*1/3^2 + 4*3^2/7^3 - 8*7^3/15^4 + 16*15^4/31^5 - 32*31^5/63^6 + 64*63^6/127^7 - 128*127^7/255^8 + 256*255^8/511^9 - 512*511^9/1023^10 + ...
where S1 = 0.84714730053329880291591114748812485885366310294051236295420...
(2) At x = -2/3: the following sums are equal
S2 = Sum_{n>=0} 3^n * (1 + 3^n)^n / (2*3^n + 1)^(n+1)
S2 = Sum_{n>=0} (-3)^n * (3^n - 1)^n / (2*3^n - 1)^(n+1).
Explicitly,
S2 = 1/3 + 3*4/7^2 + 9*10^2/19^3 + 27*28^3/55^4 + 81*82^4/163^5 + 243*244^5/487^6 + 729*730^6/1459^7 + 2187*2188^7/4375^8 + 6561*6562^8/13123^9 + 19683*19684^9/39367^10 + ...
S2 = 1 - 3*2^1/5^2 + 9*8^2/17^3 - 27*26^3/53^4 + 81*80^4/161^5 - 243*242^5/485^6 + 729*728^6/1457^7 - 2187*2186^7/4373^8 + 6561*6560^8/13121^9 - 19683*19682^9/39365^10 + ...
where S2 = 0.837457334418049175936255584889342515316005199043439291643371...
(3) At x = -1/3: the following sums are equal
S3 = Sum_{n>=0} 3^n * (2^n + 3^n)^n/(2*3^n + 2^n)^(n+1)
S3 = Sum_{n>=0} (-3)^n * (3^n - 2^n)^n/(2*3^n - 2^n)^(n+1).
Explicitly,
S3 = 1/3 + 3*5/8^2 + 9*13^2/22^3 + 27*35^3/62^4 + 81*97^4/178^5 + 243*275^5/518^6 + 729*793^6/1522^7 + 2187*2315^7/4502^8 + 6561*6817^8/13378^9 + 19683*20195^9/39878^10 + ...
S3 = 1 - 3*1/4^2 + 9*5^2/14^3 - 27*19^3/46^4 + 81*65^4/146^5 - 243*211^5/454^6 + 729*665^6/1394^7 - 2187*2059^7/4246^8 + 6561*6305^8/12866^9 - 19683*19171^9/38854^10 + ...
where S3 = 0.867357695200699139470956415922046910279987551651352471994920...

Links

Crossrefs

Cf. A302700, A122400, A302614, A302615.
Cf. A317662, A317663, A317664.

Programs

  • PARI
    {a(n) = my(A=1,o=x*O(x^n)); A = sum(m=0,n,((1+x +o)^m - 1)^m / (2 - (1+x +o)^m)^(m+1)); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (2 - (1+x)^n)^(n+1).
G.f.: Sum_{n>=0} ((1+x)^n + 1)^n / (2 + (1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317904 = 3.9561842030261697545408021818783008332999988095... and c = 0.274656497660429769528095546948772676444158... - Vaclav Kotesovec, Aug 09 2018

A323313 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + 1 - (1+x)^n)^(n+1).

Original entry on oeis.org

1, 1, 5, 42, 553, 9757, 213989, 5577285, 167819725, 5715066723, 217100774130, 9097820151376, 416870148834739, 20734141087348929, 1112567525238708108, 64067000804518722936, 3941130502347547574782, 257950869129924123671281, 17899157911139617070534480, 1312572773247815201019870104, 101429898156047011931583837037, 8238300763728888616466106146271, 701648737726537575830481016280283
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 42*x^3 + 553*x^4 + 9757*x^5 + 213989*x^6 + 5577285*x^7 + 167819725*x^8 + 5715066723*x^9 + 217100774130*x^10 + ...
such that
1 = 1/A(x) + ((1+x) - 1)/(A(x) + 1 - (1+x))^2  +  ((1+x)^2 - 1)^2/(A(x) + 1 - (1+x)^2)^3  +  ((1+x)^3 - 1)^3/(A(x) + 1 - (1+x)^3)^4  +  ((1+x)^4 - 1)^4/(A(x) + 1 - (1+x)^4)^5  +  ((1+x)^5 - 1)^5/(A(x) + 1 - (1+x)^5)^6 + ...
also,
1 = 1/(A(x) + 2)  +  (1 + (1+x))/(A(x) + 1 + (1+x))^2  +  (1 + (1+x)^2)^2/(A(x) + 1 + (1+x)^2)^3  +  (1 + (1+x)^3)^3/(A(x) + 1 + (1+x)^3)^4  +  (1 + (1+x)^4)^4/(A(x) + 1 + (1+x)^4)^5  +  (1 + (1+x)^5)^5/(A(x) + 1 + (1+x)^5)^6 + ...

Links

Crossrefs

Cf. A323311, A323573.

Programs

  • PARI
    {a(n) = my(A=[1],X=x+x*O(x^n)); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0, #A, ((1+X)^m - 1)^m / (Ser(A) + 1 - (1+X)^m)^(m+1) ) )[#A]); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + 1 - (1+x)^n)^(n+1).
(2) 1 = Sum_{n>=0} ((1+x)^n + 1)^n/(A(x) + 1 + (1+x)^n)^(n+1).
a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.546536986945124389069... - Vaclav Kotesovec, Aug 11 2021

A317351 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^(n+1) - A(x) )^n / (2 - (1+x)^n*A(x))^(n+1).

Original entry on oeis.org

1, 2, 6, 16, 154, 4584, 130464, 3816304, 123180090, 4422532004, 175136909492, 7585703878304, 356923128965592, 18139717839708536, 990827454743868120, 57910782633622271952, 3607453763547725076028, 238660376246383050751764, 16714929289459273370819900, 1235688614706272361317140840, 96170725583233854961162923028
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2018

Keywords

Comments

G.f. A(x) = G(log(1+x)), where G(x) is the e.g.f. of A317356.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 154*x^4 + 4584*x^5 + 130464*x^6 + 3816304*x^7 + 123180090*x^8 + 4422532004*x^9 + 175136909492*x^10 + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A)  +  ((1+x)^2 - A)/(2 - (1+x)*A)^2  +  ((1+x)^3 - A)^2/(2 - (1+x)^2*A)^3  +  ((1+x)^4 - A)^3/(2 - (1+x)^3*A)^4  +  ((1+x)^5 - A)^4/(2 - (1+x)^4*A)^5  +  ((1+x)^6 - A)^5/(2 - (1+x)^5*A)^6 + ...
Also,
A(x) = 1/(2 + A)  +  ((1+x)^2 + A)/(2 + (1+x)*A)^2  +  ((1+x)^3 + A)^2/(2 + (1+x)^2*A)^3  +  ((1+x)^4 + A)^3/(2 + (1+x)^3*A)^4  +  ((1+x)^5 + A)^4/(2 + (1+x)^4*A)^5  +  ((1+x)^6 + A)^5/(2 + (1+x)^5*A)^6 + ...

Links

Crossrefs

Cf. A317350, A317356.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A = Vec( sum(m=0,#A, ( (1+x)^(m+1) - Ser(A) )^m  / (2 - (1+x)^m*Ser(A))^(m+1) ) ) ); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( (1+x)^(n+1) - A(x) )^n / (2 - (1+x)^n*A(x))^(n+1),
(2) A(x) = Sum_{n>=0} ( (1+x)^(n+1) + A(x) )^n / (2 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317904 = 3.956184203026... and c = 0.23137523927... - Vaclav Kotesovec, Aug 07 2018

A317356 E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1).

Original entry on oeis.org

1, 2, 14, 134, 4358, 589622, 102434534, 21285122294, 5530748479718, 1792785367579382, 711595226383338854, 339665400624638782454, 192071493764203628322278, 127053485326157331378577142, 97253813187878484942034153574, 85330814329687863076988482842614, 85104598195236153766017309663096038
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2018

Keywords

Comments

E.g.f. A(x) = G(exp(x) - 1), where G(x) is the g.f. of A317351.

Examples

			E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 134*x^3/3! + 4358*x^4/4! + 589622*x^5/5! + 102434534*x^6/6! + 21285122294*x^7/7! + 5530748479718*x^8/8! + 1792785367579382*x^9/9! + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A)  +  (exp(2*x) - A)/(2 - exp(x)*A)^2  +  (exp(3*x) - A)^2/(2 - exp(2*x)*A)^3  +  (exp(4*x) - A)^3/(2 - exp(3*x)*A)^4  +  (exp(5*x) - A)^4/(2 - exp(4*x)*A)^5  +  (exp(6*x) - A)^5/(2 - exp(5*x)*A)^6 + ...
Also,
A(x) = 1/(2 + A)  +  (exp(2*x) + A)/(2 + exp(x)*A)^2  +  (exp(3*x) + A)^2/(2 + exp(2*x)*A)^3  +  (exp(4*x) + A)^3/(2 + exp(3*x)*A)^4  +  (exp(5*x) + A)^4/(2 + exp(4*x)*A)^5  +  (exp(6*x) + A)^5/(2 + exp(5*x)*A)^6 + ...

Links

Crossrefs

Cf. A317356, A317351.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( exp((m+1)*x +x*O(x^#A)) - Ser(A) )^m  / (2 - exp(m*x +x*O(x^#A))*Ser(A))^(m+1) ) ) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1),
(2) A(x) = Sum_{n>=0} ( exp((n+1)*x) + A(x) )^n / (2 + exp(n*x)*A(x))^(n+1).
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.2625457134... - Vaclav Kotesovec, Aug 10 2018

A322737 G.f. satisfies: A(x) = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n / (2 - A(x)/(1-x)^n)^(n+1).

Original entry on oeis.org

1, 1, 3, 17, 243, 5041, 122793, 3433557, 108824679, 3857180303, 151189425233, 6495604450659, 303671019221745, 15353507145898735, 835092643075565163, 48637547540923032151, 3020890094905581400107, 199356631125403317760803, 13932407051414083995444277, 1028080194901048673942405547, 79883891921410823861579965753
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 243*x^4 + 5041*x^5 + 122793*x^6 + 3433557*x^7 + 108824679*x^8 + 3857180303*x^9 + 151189425233*x^10 + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A)  +  (1/(1-x) - A)/(2 - A/(1-x))^2  +  (1/(1-x)^2 - A)^2/(2 - A/(1-x)^2)^3  +  (1/(1-x)^3 - A)^3/(2 - A/(1-x)^3)^4  +  (1/(1-x)^4 - A)^4/(2 - A/(1-x)^4)^5  +  (1/(1-x)^5 - A)^5/(2 - A/(1-x)^5)^6 + ...
Also,
A(x) = 1/(2 + A)  +  (1/(1-x) + A)/(2 + A/(1-x))^2  +  (1/(1-x)^2 + A)^2/(2 + A/(1-x)^2)^3  +  (1/(1-x)^3 + A)^3/(2 + A/(1-x)^3)^4  +  (1/(1-x)^4 + A)^4/(2 + A/(1-x)^4)^5  +  (1/(1-x)^5 + A)^5/(2 + A/(1-x)^5)^6 + ...

Links

Crossrefs

Cf. A317350.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( 1/(1-x)^m - Ser(A) )^m  / (2 - Ser(A)/(1-x)^m)^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n / (2 - A(x)/(1-x)^n)^(n+1),
(2) A(x) = Sum_{n>=0} ( 1/(1-x)^n + A(x) )^n / (2 + A(x)/(1-x)^n)^(n+1).
a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.47061136383707... - Vaclav Kotesovec, Aug 11 2021

A302700 E.g.f.: Sum_{n>=0} (exp(n*x) + 1)^n / (2 + exp(n*x))^(n+1).

Original entry on oeis.org

1, 1, 13, 385, 21325, 1898401, 247841293, 44611568065, 10589093387725, 3204648461107681, 1204384753185644173, 550313048077989740545, 300436578515074737333325, 193139598305033634851120161, 144410707207961955130172624653, 124258444226932649355925701301825, 121911793079671988588136925596434125, 135284324089583933279712302959420767841
Offset: 0

Views

Author

Paul D. Hanna, Apr 14 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 13*x^2/2! + 385*x^3/3! + 21325*x^4/4! + 1898401*x^5/5! + 247841293*x^6/6! + 44611568065*x^7/7! + 10589093387725*x^8/8! + 3204648461107681*x^9/9! + ...
such that
A(x) = 1/3 + (exp(x)+1)/(2+exp(x))^2 + (exp(2*x)+1)^2/(2+exp(2*x))^3 + (exp(3*x)+1)^3/(2+exp(3*x))^4 + (exp(4*x)+1)^4/(2+exp(4*x))^5 + (exp(5*x)+1)^5/(2+exp(5*x))^6 + (exp(6*x)+1)^6/(2+exp(6*x))^7 + ...
Also,
A(x) = 1 + (exp(x)-1)/(2-exp(x))^2 + (exp(2*x)-1)^2/(2-exp(2*x))^3 + (exp(3*x)-1)^3/(2-exp(3*x))^4 + (exp(4*x)-1)^4/(2-exp(4*x))^5 + (exp(5*x)-1)^5/(2-exp(5*x))^6 + (exp(6*x)-1)^6/(2-exp(6*x))^7 + ...

Links

Crossrefs

Cf. A302598.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(E^(k*x) - 1)^k / (2 - E^(k*x))^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 11 2018 *)
  • PARI
    {a(n) = my(A=1); A = sum(m=0,n+1, (exp(m*x + x*O(x^n)) - 1)^m / (2 - exp(m*x + x*O(x^n)))^(m+1) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! equals:
(1) Sum_{n>=0} (exp(n*x) + 1)^n / (2 + exp(n*x))^(n+1).
(2) Sum_{n>=0} (exp(n*x) - 1)^n / (2 - exp(n*x))^(n+1).
(3) Sum_{n>=0} 2^n*exp(n^2*x/2)*cosh(n*x/2)^n/(1 + 2*exp(n*x/2)*cosh(n*x/2))^(n+1).
(4) Sum_{n>=0} 2^n*exp(n^2*x/2)*sinh(n*x/2)^n/(1 - 2*exp(n*x/2)*sinh(n*x/2))^(n+1).
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.31165774853025500197969363638844... - Vaclav Kotesovec, Aug 10 2018

A317355 E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp(n*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1).

Original entry on oeis.org

1, 1, 5, 85, 5261, 549061, 79707245, 15531175045, 3926159465261, 1249497583485061, 488841071584907885, 230674363972514998405, 129251110556658394610861, 84870052450743141454787461, 64574784437643167984687238125, 56377769340759003121860283852165, 55996026841326090728124344073814061
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2018

Keywords

Comments

E.g.f. A(x) = G(exp(x) - 1), where G(x) is the g.f. of A317350.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 85*x^3/3! + 5261*x^4/4! + 549061*x^5/5! + 79707245*x^6/6! + 15531175045*x^7/7! + 3926159465261*x^8/8! + 1249497583485061*x^9/9! + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A)  +  (exp(x) - A)/(2 - exp(x)*A)^2  +  (exp(2*x) - A)^2/(2 - exp(2*x)*A)^3  +  (exp(3*x) - A)^3/(2 - exp(3*x)*A)^4  +  (exp(4*x) - A)^4/(2 - exp(4*x)*A)^5  +  (exp(5*x) - A)^5/(2 - exp(5*x)*A)^6 + ...
Also,
A(x) = 1/(2 + A)  +  (exp(x) + A)/(2 + exp(x)*A)^2  +  (exp(2*x) + A)^2/(2 + exp(2*x)*A)^3  +  (exp(3*x) + A)^3/(2 + exp(3*x)*A)^4  +  (exp(4*x) + A)^4/(2 + exp(4*x)*A)^5  +  (exp(5*x) + A)^5/(2 + exp(5*x)*A)^6 + ...

Links

Crossrefs

Cf. A317356, A317350.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( exp(m*x +x*O(x^#A)) - Ser(A) )^m  / (2 - exp(m*x +x*O(x^#A))*Ser(A))^(m+1) ) ) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( exp(n*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1),
(2) A(x) = Sum_{n>=0} ( exp(n*x) + A(x) )^n / (2 + exp(n*x)*A(x))^(n+1).
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.16545672527... - Vaclav Kotesovec, Aug 10 2018

A322735 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x)^(1/2) )^n / ( 2 - (1+x)^n * A(x)^(1/2) )^(n+1).

Original entry on oeis.org

1, 1, 4, 32, 424, 7696, 173442, 4619266, 141315896, 4874012942, 186981188532, 7896318230898, 364045464940596, 18196879341802488, 980406767669688312, 56648325010279262864, 3494752526532046751322, 229295129566323954429582, 15944415062268028208782178, 1171388932048172852048806000, 90667183883120180538001042398
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2019

Keywords

Comments

It is remarkable that the g.f. should consist entirely of integer coefficients.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 424*x^4 + 7696*x^5 + 173442*x^6 + 4619266*x^7 + 141315896*x^8 + 4874012942*x^9 + 186981188532*x^10 + ...
such that A(x) and B = A(x)^(1/2) satisfy
A(x) = 1/(2 - B)  +  ((1+x) - B)/(2 - (1+x)*B)^2  +  ((1+x)^2 - B)^2/(2 - (1+x)^2*B)^3  +  ((1+x)^3 - B)^3/(2 - (1+x)^3*B)^4  +  ((1+x)^4 - B)^4/(2 - (1+x)^4*B)^5  +  ((1+x)^5 - B)^5/(2 - (1+x)^5*B)^6 + ...
also,
A(x) = 1/(2 + B)  +  ((1+x) + B)/(2 + (1+x)*B)^2  +  ((1+x)^2 + B)^2/(2 + (1+x)^2*B)^3  +  ((1+x)^3 + B)^3/(2 + (1+x)^3*B)^4  +  ((1+x)^4 + B)^4/(2 + (1+x)^4*B)^5  +  ((1+x)^5 + B)^5/(2 + (1+x)^5*B)^6 + ...
Notice that A(x)^(1/2) is not an integer series, but instead begins
A(x)^(1/2) = 1 + 2*(x/4) + 30*(x/4)^2 + 964*(x/4)^3 + 51894*(x/4)^4 + 3807644*(x/4)^5 + 345572460*(x/4)^6 + 36985627016*(x/4)^7 + 4541283789862*(x/4)^8 + 628123762214444*(x/4)^9 + 96578670976842436*(x/4)^10 + ...
thus, given the definition, it is remarkable that A(x) should be an integer series.

Links

Crossrefs

Cf. A317350, A322737.

Programs

  • PARI
    {a(n) = my(A=[1,1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A)^(1/2) )^m  / (2 - (1+x)^m*Ser(A)^(1/2))^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) along with B(x) = A(x)^(1/2) satisfy:
(1) A(x) = Sum_{n>=0} ( (1+x)^n - B(x) )^n / ( 2 - (1+x)^n*B(x) )^(n+1),
(2) A(x) = Sum_{n>=0} ( (1+x)^n + B(x) )^n / ( 2 + (1+x)^n*B(x) )^(n+1).
a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.501629489631036... - Vaclav Kotesovec, Jul 03 2025

A323311 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (exp(n*x) - 1)^n/(A(x) + 1 - exp(n*x))^(n+1).

Original entry on oeis.org

1, 1, 11, 283, 14855, 1310011, 172520351, 31513669363, 7595793146855, 2330879613371851, 886383762411615791, 408963256168949033443, 225040270250903527024055, 145601653678200482159541691, 109437844707983885536850408831, 94572173789825201408460630621523, 93118733370917669491764504635160455, 103644400582305503214140030821130959531, 129490690058782610512772741408027302955471, 180464581077334737195826400036356606725361603
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 11*x^2/2! + 283*x^3/3! + 14855*x^4/4! + 1310011*x^5/5! + 172520351*x^6/6! + 31513669363*x^7/7! + 7595793146855*x^8/8! + 2330879613371851*x^9/9! +  + 886383762411615791*x^10/10! + ...
such that
1 = 1/A(x)  +  (exp(x) - 1)/(A(x) + 1 - exp(x))^2  +  (exp(2*x) - 1)^2/(A(x) + 1 - exp(2*x))^3  +  (exp(3*x) - 1)^3/(A(x) + 1 - exp(3*x))^4  +  (exp(4*x) - 1)^4/(A(x) + 1 - exp(4*x))^5  +  (exp(5*x) - 1)^5/(A(x) + 1 - exp(5*x))^6 + ...
also,
1 = 1/(A(x) + 2)  +  (exp(x) + 1)/(A(x) + 1 + exp(x))^2  +  (exp(2*x) + 1)^2/(A(x) + 1 + exp(2*x))^3  +  (exp(3*x) + 1)^3/(A(x) + 1 + exp(3*x))^4  +  (exp(4*x) + 1)^4/(A(x) + 1 + exp(4*x))^5  +  (exp(5*x) + 1)^5/(A(x) + 1 + exp(5*x))^6 + ...
RELATED SERIES.
log(A(x)) = x + 10*x^2/2! + 252*x^3/3! + 13486*x^4/4! + 1213260*x^5/5! + 162204670*x^6/6! + 29956649772*x^7/7! + 7279075598686*x^8/8! + 2247264600871500*x^9/9! + ...

Links

Crossrefs

Cf. A323313.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0, #A, (exp(m*x +x*O(x^n)) - 1)^m / (Ser(A) + 1 - exp(m*x +x*O(x^n)))^(m+1) ) )[#A]); n!*A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (exp(n*x) - 1)^n/(A(x) + 1 - exp(n*x))^(n+1).
(2) 1 = Sum_{n>=0} (exp(n*x) + 1)^n/(A(x) + 1 + exp(n*x))^(n+1).
a(n) ~ c * A317904^n * n^(2*n + 1/2) / exp(2*n), where c = 1.5545244013... - Vaclav Kotesovec, Aug 11 2021
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