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 1-4 of 4 results.

A317349 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n = 1.

Original entry on oeis.org

1, 1, 2, 7, 42, 372, 4269, 59047, 946557, 17175289, 347208299, 7730688884, 187911183701, 4951155672353, 140575561645293, 4279249948000903, 139050095246322895, 4804391579357016747, 175902340755219278039, 6803436418471129704925, 277202774381386656583959, 11868116969794805874111831
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 42*x^4 + 372*x^5 + 4269*x^6 + 59047*x^7 + 946557*x^8 + 17175289*x^9 + 347208299*x^10 + ...
such that
1 = 1  +  (1/A(x) - (1-x))  +  (1/A(x) - (1-x)^2)^2  +  (1/A(x) - (1-x)^3)^3  +  (1/A(x) - (1-x)^4)^4  +  (1/A(x) - (1-x)^5)^5  +  (1/A(x) - (1-x)^6)^6  +  (1/A(x) - (1-x)^7)^7  +  (1/A(x) - (1-x)^8)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - (1-x)^2)  +  (1/A(x) - (1-x)^3)^2  +  (1/A(x) - (1-x)^4)^3  +  (1/A(x) - (1-x)^5)^4  +  (1/A(x) - (1-x)^6)^5  +  (1/A(x) - (1-x)^7)^6  +  (1/A(x) - (1-x)^8)^7  +  (1/A(x) - (1-x)^9)^8  + ...

Links

Crossrefs

Cf. A317339, A317348, A317340, A317666, A317667 A317668.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( 1/Ser(A) - (1-x)^(m+1) )^m ) )[#A]/2 ); A[n+1]}
for(n=0,25, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n * (1-x)^(n+1).
(4) A(x)^2 = 2*A(x) * [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+1) )^n ] - [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+2) )^n ].
(5) A(x) = [ Sum_{n>=1} n*(n+1)/2 * (1-x)^(n+1) * ( 1/Ser(A) - (1-x)^(n+1) )^(n-1) ] / [ Sum_{n>=1} n^2 * (1-x)^n * ( 1/Ser(A) - (1-x)^n )^(n-1) ].
a(n) ~ 2^(log(2)/2 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 06 2018

A317801 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n) )^n = 1.

Original entry on oeis.org

1, 2, 5, 36, 465, 8504, 196900, 5448960, 174282930, 6304990774, 254130115519, 11285072891720, 547300883009420, 28782595694322682, 1631647871872693931, 99196878522983084624, 6438935162966517263673, 444501575692314862825620, 32520919561355194120182078, 2513781290973908970634293260, 204713526722520414595009119193
Offset: 0

Views

Author

Paul D. Hanna, Aug 12 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 36*x^3 + 465*x^4 + 8504*x^5 + 196900*x^6 + 5448960*x^7 + 174282930*x^8 + 6304990774*x^9 + 254130115519*x^10 + ...
such that
1 = 1  +  (1/A(x) - 1/(1+x)^2)  +  (1/A(x) - 1/(1+x)^4)^2  +  (1/A(x) - 1/(1+x)^6)^3  +  (1/A(x) - 1/(1+x)^8)^4  +  (1/A(x) - 1/(1+x)^10)^5  +  (1/A(x) - 1/(1+x)^12)^6  +  (1/A(x) - 1/(1+x)^14)^7  +  (1/A(x) - 1/(1+x)^16)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - 1/(1+x)^4)  +  (1/A(x) - 1/(1+x)^6)^2  +  (1/A(x) - 1/(1+x)^8)^3  +  (1/A(x) - 1/(1+x)^10)^4  +  (1/A(x) - 1/(1+x)^12)^5  +  (1/A(x) - 1/(1+x)^14)^6  +  (1/A(x) - 1/(1+x)^16)^7  +  (1/A(x) - 1/(1+x)^18)^8  + ...
RELATED SERIES.
The series B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+1) )^n begins
B(x) = 1 + x + 2*x^2 + 15*x^3 + 193*x^4 + 3523*x^5 + 81497*x^6 + 2254695*x^7 + 72114516*x^8 + 2609080565*x^9 + 105173092838*x^10 + ...
restated,
B(x) = 1  +  (1/A(x) - 1/(1+x)^3)  +  (1/A(x) - 1/(1+x)^5)^2  +  (1/A(x) - 1/(1+x)^7)^3  +  (1/A(x) - 1/(1+x)^9)^4  +  (1/A(x) - 1/(1+x)^11)^5  +  (1/A(x) - 1/(1+x)^13)^6  +  (1/A(x) - 1/(1+x)^15)^7  +  (1/A(x) - 1/(1+x)^17)^8  + ...
which can also be written
B(x) = 1/(1+x)  +  (1/A(x) - 1/(1+x)^4)/(1+x)^2  +  (1/A(x) - 1/(1+x)^6)^2/(1+x)^3  +  (1/A(x) - 1/(1+x)^8)^3/(1+x)^4  +  (1/A(x) - 1/(1+x)^10)^4/(1+x)^5  +  (1/A(x) - 1/(1+x)^12)^5/(1+x)^6  +  (1/A(x) - 1/(1+x)^14)^6/(1+x)^7  +  (1/A(x) - 1/(1+x)^16)^7/(1+x)^8  +  (1/A(x) - 1/(1+x)^18)^8/(1+x)^9  + ...
Compare the above to
1 = 1/(1+x)^2  +  (1/A(x) - 1/(1+x)^4)/(1+x)^4  +  (1/A(x) - 1/(1+x)^6)^2/(1+x)^6  +  (1/A(x) - 1/(1+x)^8)^3/(1+x)^8  +  (1/A(x) - 1/(1+x)^10)^4/(1+x)^10  +  (1/A(x) - 1/(1+x)^12)^5/(1+x)^12  +  (1/A(x) - 1/(1+x)^14)^6/(1+x)^14  +  (1/A(x) - 1/(1+x)^16)^7/(1+x)^16  +  (1/A(x) - 1/(1+x)^18)^8/(1+x)^18  + ...

Links

Crossrefs

Cf. A317339, A317802, A317803, A317666.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - 1/(1+x +x*O(x^#A))^(2*m+2) )^m ) )[#A]/2 ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n) )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+2) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+2) )^n / (1+x)^(2*n+2).
(4) Let B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+1) )^n ,
then B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+2) )^n / (1+x)^(n+1).
a(n) ~ 2^(n - log(2)/4 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018

A317667 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n) )^n = 1.

Original entry on oeis.org

1, 3, 15, 154, 2865, 77532, 2684504, 111490839, 5357828286, 291299582266, 17643988446921, 1177175235308976, 85754781272021397, 6772714984220704506, 576470959628636447748, 52613628461306161087953, 5126338275850981999654524, 531146069930403178373329794, 58319563977901655667747310206, 6764879932357508722274792757285
Offset: 0

Views

Author

Paul D. Hanna, Aug 12 2018

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 154*x^3 + 2865*x^4 + 77532*x^5 + 2684504*x^6 + 111490839*x^7 + 5357828286*x^8 + 291299582266*x^9 + 17643988446921*x^10 + ...
such that
1 = 1  +  (1/A(x) - (1-x)^3)  +  (1/A(x) - (1-x)^6)^2  +  (1/A(x) - (1-x)^9)^3  +  (1/A(x) - (1-x)^12)^4  +  (1/A(x) - (1-x)^15)^5  +  (1/A(x) - (1-x)^18)^6  +  (1/A(x) - (1-x)^21)^7  +  (1/A(x) - (1-x)^24)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - (1-x)^6)  +  (1/A(x) - (1-x)^9)^2  +  (1/A(x) - (1-x)^12)^3  +  (1/A(x) - (1-x)^15)^4  +  (1/A(x) - (1-x)^18)^5  +  (1/A(x) - (1-x)^21)^6  +  (1/A(x) - (1-x)^24)^7  +  (1/A(x) - (1-x)^27)^8  + ...
RELATED SERIES.
(1) The series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+1) )^n begins
B(x) = 1 + x + 4*x^2 + 40*x^3 + 743*x^4 + 20073*x^5 + 694477*x^6 + 28841790*x^7 + 1386441234*x^8 + 75408643207*x^9 + 4569235921823*x^10 + ...
restated,
B(x) = 1  +  (1/A(x) - (1-x)^4)  +  (1/A(x) - (1-x)^7)^2  +  (1/A(x) - (1-x)^10)^3  +  (1/A(x) - (1-x)^13)^4  +  (1/A(x) - (1-x)^16)^5  +  (1/A(x) - (1-x)^19)^6  +  (1/A(x) - (1-x)^22)^7  +  (1/A(x) - (1-x)^25)^8  + ...
which can also be written
B(x) = (1-x)^2  +  (1/A(x) - (1-x)^6)*(1-x)^4  +  (1/A(x) - (1-x)^9)^2*(1-x)^6  +  (1/A(x) - (1-x)^12)^3*(1-x)^8  +  (1/A(x) - (1-x)^15)^4*(1-x)^10  +  (1/A(x) - (1-x)^18)^5*(1-x)^12  +  (1/A(x) - (1-x)^21)^6*(1-x)^14  +  (1/A(x) - (1-x)^24)^7*(1-x)^16  +  (1/A(x) - (1-x)^27)^8*(1-x)^18  + ...
...
(2) The series C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+2) )^n begins
C(x) = 1 + 2*x + 9*x^2 + 91*x^3 + 1690*x^4 + 45661*x^5 + 1579367*x^6 + 65559850*x^7 + 3149821447*x^8 + 171233732325*x^9 + 10371022987322*x^10 + ...
restated,
C(x) = 1  +  (1/A(x) - (1-x)^5)  +  (1/A(x) - (1-x)^8)^2  +  (1/A(x) - (1-x)^11)^3  +  (1/A(x) - (1-x)^14)^4  +  (1/A(x) - (1-x)^17)^5  +  (1/A(x) - (1-x)^20)^6  +  (1/A(x) - (1-x)^23)^7  +  (1/A(x) - (1-x)^26)^8  + ...
which can also be written
C(x) = (1-x)  +  (1/A(x) - (1-x)^6)*(1-x)^2  +  (1/A(x) - (1-x)^9)^2*(1-x)^3  +  (1/A(x) - (1-x)^12)^3*(1-x)^4  +  (1/A(x) - (1-x)^15)^4*(1-x)^5  +  (1/A(x) - (1-x)^18)^5*(1-x)^6  +  (1/A(x) - (1-x)^21)^6*(1-x)^7  +  (1/A(x) - (1-x)^24)^7*(1-x)^8  +  (1/A(x) - (1-x)^27)^8*(1-x)^9  + ...
...
Compare the above series to
1 = (1-x)^3  +  (1/A(x) - (1-x)^6)*(1-x)^6  +  (1/A(x) - (1-x)^9)^2*(1-x)^9  +  (1/A(x) - (1-x)^12)^3*(1-x)^12  +  (1/A(x) - (1-x)^15)^4*(1-x)^15  +  (1/A(x) - (1-x)^18)^5*(1-x)^18  +  (1/A(x) - (1-x)^21)^6*(1-x)^21  +  (1/A(x) - (1-x)^24)^7*(1-x)^24  +  (1/A(x) - (1-x)^27)^8*(1-x)^27  + ...

Links

Crossrefs

Cf. A317349, A317666, A317668, A317802.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - (1-x)^(3*m+3) )^m ) )[#A]/2 ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n) )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n * (1-x)^(3*n+3).
(4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+1) )^n,
then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n * (1-x)^(2*n+2).
(5) Let C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+2) )^n,
then C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n * (1-x)^(n+1).
a(n) ~ 2^(log(2)/6 - 5/2) * 3^n * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018

A317668 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n) )^n = 1.

Original entry on oeis.org

1, 4, 26, 356, 8871, 320672, 14811200, 820185072, 52546341422, 3808527303300, 307523461730866, 27352330591164308, 2656394433081980649, 279696497208771609120, 31739466678890197201328, 3862114024795578127697248, 501700135604304149492422266, 69305144023051764776753873168, 10145743117833906529065611237208, 1569100081969097895595627120200512
Offset: 0

Views

Author

Paul D. Hanna, Aug 12 2018

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 356*x^3 + 8871*x^4 + 320672*x^5 + 14811200*x^6 + 820185072*x^7 + 52546341422*x^8 + 3808527303300*x^9 + 307523461730866*x^10 + ...
such that
1 = 1  +  (1/A(x) - (1-x)^4)  +  (1/A(x) - (1-x)^8)^2  +  (1/A(x) - (1-x)^12)^3  +  (1/A(x) - (1-x)^16)^4  +  (1/A(x) - (1-x)^20)^5  +  (1/A(x) - (1-x)^24)^6  +  (1/A(x) - (1-x)^28)^7  +  (1/A(x) - (1-x)^32)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - (1-x)^8)  +  (1/A(x) - (1-x)^12)^2  +  (1/A(x) - (1-x)^16)^3  +  (1/A(x) - (1-x)^20)^4  +  (1/A(x) - (1-x)^24)^5  +  (1/A(x) - (1-x)^28)^6  +  (1/A(x) - (1-x)^32)^7  +  (1/A(x) - (1-x)^36)^8  + ...
RELATED SERIES.
(1) The series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+1) )^n begins
B(x) = 1 + x + 5*x^2 + 67*x^3 + 1669*x^4 + 60246*x^5 + 2781335*x^6 + 154062232*x^7 + 9875799121*x^8 + 716231200582*x^9 + 57865799711347*x^10 + ...
also given by B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(3*n+3).
(2) The series C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+2) )^n begins
C(x) = 1 + 2*x + 11*x^2 + 148*x^3 + 3683*x^4 + 132888*x^5 + 6131332*x^6 + 339397944*x^7 + 21742672693*x^8 + 1575995237188*x^9 + 127268039660042*x^10 + ...
also given by C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(2*n+2).
(3) The series D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+3) )^n begins
D(x) = 1 + 3*x + 18*x^2 + 244*x^3 + 6073*x^4 + 219238*x^5 + 10117351*x^6 + 560000464*x^7 + 35868610134*x^8 + 2599382401532*x^9 + 209871544727484*x^10 + ...
also given by D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(n+1).

Links

Crossrefs

Cf. A317349, A317666, A317667, A317803.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - (1-x)^(4*m+4) )^m ) )[#A]/2 ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n) )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(4*n+4).
(4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+1) )^n,
then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(3*n+3).
(5) Let C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+2) )^n,
then C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(2*n+2).
(6) Let D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+3) )^n,
then D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(n+1).
a(n) ~ 2^(2*n + log(2)/8 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018
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