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 1-5 of 5 results.

A317339 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^n )^n = 1.

Original entry on oeis.org

1, 1, 1, 4, 26, 239, 2768, 38267, 611193, 11040954, 222241117, 4929304517, 119423079917, 3137864557135, 88884310756274, 2700439386780586, 87603920737623984, 3022626187893726774, 110534722263602544357, 4270777627515614565004, 173854104446646589718022, 7437462737558953036993295
Offset: 0

Views

Author

Paul D. Hanna, Aug 10 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 26*x^4 + 239*x^5 + 2768*x^6 + 38267*x^7 + 611193*x^8 + 11040954*x^9 + 222241117*x^10 + ...
such that
1 = 1  +  (1/A(x) - 1/(1+x))  +  (1/A(x) - 1/(1+x)^2)^2  +  (1/A(x) - 1/(1+x)^3)^3  +  (1/A(x) - 1/(1+x)^4)^4  +  (1/A(x) - 1/(1+x)^5)^5  +  (1/A(x) - 1/(1+x)^6)^6  +  (1/A(x) - 1/(1+x)^7)^7  +  (1/A(x) - 1/(1+x)^8)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - 1/(1+x)^2)  +  (1/A(x) - 1/(1+x)^3)^2  +  (1/A(x) - 1/(1+x)^4)^3  +  (1/A(x) - 1/(1+x)^5)^4  +  (1/A(x) - 1/(1+x)^6)^5  +  (1/A(x) - 1/(1+x)^7)^6  +  (1/A(x) - 1/(1+x)^8)^7  +  (1/A(x) - 1/(1+x)^9)^8  + ...

Links

Crossrefs

Cf. A317801, A317802, A317803, A317349.

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)^(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/(1+x)^n )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(n+1) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(n+1) )^n / (1+x)^(n+1).
a(n) ~ n^n / (2^(log(2)/2 + 5/2) * sqrt(1-log(2)) * exp(n) * (log(2))^(2*n + 1)). - Vaclav Kotesovec, Aug 12 2018

A317666 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n) )^n = 1.

Original entry on oeis.org

1, 2, 7, 48, 590, 10602, 244457, 6767792, 216875258, 7863473864, 317632851912, 14132208327052, 686514289288897, 36154193924315170, 2051928741855927465, 124870207134047889232, 8112089716821244526285, 560396754826502247713090, 41024663835523296400398275, 3172738829903313189522259140, 258493327059457440608140711531
Offset: 0

Views

Author

Paul D. Hanna, Aug 12 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 48*x^3 + 590*x^4 + 10602*x^5 + 244457*x^6 + 6767792*x^7 + 216875258*x^8 + 7863473864*x^9 + 317632851912*x^10 + ...
such that
1 = 1  +  (1/A(x) - (1-x)^2)  +  (1/A(x) - (1-x)^4)^2  +  (1/A(x) - (1-x)^6)^3  +  (1/A(x) - (1-x)^8)^4  +  (1/A(x) - (1-x)^10)^5  +  (1/A(x) - (1-x)^12)^6  +  (1/A(x) - (1-x)^14)^7  +  (1/A(x) - (1-x)^16)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - (1-x)^4)  +  (1/A(x) - (1-x)^6)^2  +  (1/A(x) - (1-x)^8)^3  +  (1/A(x) - (1-x)^10)^4  +  (1/A(x) - (1-x)^12)^5  +  (1/A(x) - (1-x)^14)^6  +  (1/A(x) - (1-x)^16)^7  +  (1/A(x) - (1-x)^18)^8  + ...
RELATED SERIES.
The related series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+1) )^n begins
B(x) = 1 + x + 3*x^2 + 20*x^3 + 245*x^4 + 4394*x^5 + 101203*x^6 + 2800620*x^7 + 89739208*x^8 + 3253949840*x^9 + 131451064170*x^10 + ...
restated,
B(x) = 1  +  (1/A(x) - (1-x)^3)  +  (1/A(x) - (1-x)^5)^2  +  (1/A(x) - (1-x)^7)^3  +  (1/A(x) - (1-x)^9)^4  +  (1/A(x) - (1-x)^11)^5  +  (1/A(x) - (1-x)^13)^6  +  (1/A(x) - (1-x)^15)^7  +  (1/A(x) - (1-x)^17)^8  + ...
which also equals
B(x) = (1-x)  +  (1/A(x) - (1-x)^4)*(1-x)^2  +  (1/A(x) - (1-x)^6)^2*(1-x)^3  +  (1/A(x) - (1-x)^8)^3*(1-x)^4  +  (1/A(x) - (1-x)^10)^4*(1-x)^5  +  (1/A(x) - (1-x)^12)^5*(1-x)^6  +  (1/A(x) - (1-x)^14)^6*(1-x)^7  +  (1/A(x) - (1-x)^16)^7*(1-x)^8  +  (1/A(x) - (1-x)^18)^8*(1-x)^9  + ...
Compare the above to
1 = (1-x)^2  +  (1/A(x) - (1-x)^4)*(1-x)^4  +  (1/A(x) - (1-x)^6)^2*(1-x)^6  +  (1/A(x) - (1-x)^8)^3*(1-x)^8  +  (1/A(x) - (1-x)^10)^4*(1-x)^10  +  (1/A(x) - (1-x)^12)^5*(1-x)^12  +  (1/A(x) - (1-x)^14)^6*(1-x)^14  +  (1/A(x) - (1-x)^16)^7*(1-x)^16  +  (1/A(x) - (1-x)^18)^8*(1-x)^18  + ...

Links

Crossrefs

Cf. A317349, A317667, A317668, A317801.

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)^(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-x)^(2*n) )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+2) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+2) )^n * (1-x)^(2*n+2).
(4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+1) )^n ,
then B(x) = Sum_{n>=0} ( 1/A(x) - (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

A317348 E.g.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - exp(-n*x) )^n = 1.

Original entry on oeis.org

1, 1, 3, 31, 783, 35551, 2465943, 238958791, 30604867023, 4988281843471, 1006426188747783, 246050857141536151, 71658459729884788863, 24512979124556543501791, 9733113984959380709677623, 4440214540533789234079579111, 2306721251730615059447461056303, 1354037785009235729190621178158511
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 783*x^4/4! + 35551*x^5/5! + 2465943*x^6/6! + 238958791*x^7/7! + 30604867023*x^8/8! + 4988281843471*x^9/9! + ...
such that
1 = 1  +  (1/A(x) - exp(-x))  +  (1/A(x) - exp(-2*x))^2  +  (1/A(x) - exp(-3*x))^3  +  (1/A(x) - exp(-4*x))^4  +  (1/A(x) - exp(-5*x))^5  +  (1/A(x) - exp(-6*x))^6  +  (1/A(x) - exp(-7*x))^7  +  (1/A(x) - exp(-8*x))^8  + ...
Also,
A(x) = 1  +  (1/A(x) - exp(-2*x))  +  (1/A(x) - exp(-3*x))^2  +  (1/A(x) - exp(-4*x))^3  +  (1/A(x) - exp(-5*x))^4  +  (1/A(x) - exp(-6*x))^5  +  (1/A(x) - exp(-7*x))^6  +  (1/A(x) - exp(-8*x))^7  +  (1/A(x) - exp(-9*x))^8  + ...

Links

Crossrefs

Cf. A317349.

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) - exp(-(m+1)*x +x*O(x^#A)) )^m ) )[#A]/2 ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - exp(-n*x) )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - exp(-(n+1)*x) )^n.
(3) 1 = Sum_{n>=0} exp(-(n+1)*x) * ( 1/A(x) - exp(-(n+1)*x) )^n.
(4) A(x)^2 = 2*A(x) * [ Sum_{n>=0} (n+1) * ( 1/A(x) - exp(-(n+1)*x) )^n ] - [ Sum_{n>=0} (n+1) * ( 1/A(x) - exp(-(n+2)*x) )^n ].
(5) A(x) = [ Sum_{n>=1} n*(n+1)/2 * exp(-(n+1)*x) * ( 1/Ser(A) - exp(-(n+1)*x) )^(n-1) ] / [ Sum_{n>=1} n^2 * exp(-n*x) * ( 1/Ser(A) - exp(-n*x) )^(n-1) ].
a(n) ~ sqrt(Pi) * n^(2*n + 1/2) / (4*sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 10 2018
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