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 1-8 of 8 results.

A317904 Decimal expansion of a constant related to the asymptotics of A302598.

Original entry on oeis.org

3, 9, 5, 6, 1, 8, 4, 2, 0, 3, 0, 2, 6, 1, 6, 9, 7, 5, 4, 5, 4, 0, 8, 0, 2, 1, 8, 1, 8, 7, 8, 3, 0, 0, 8, 3, 3, 2, 9, 9, 9, 9, 8, 8, 0, 9, 5, 2, 5, 5, 5, 4, 0, 9, 8, 1, 6, 4, 9, 6, 2, 1, 1, 3, 0, 9, 3, 2, 8, 8, 5, 1, 3, 1, 4, 6, 2, 5, 2, 1, 2, 3, 0, 3, 1, 1, 5, 8, 9, 4, 8, 8, 4, 1, 6, 4, 0, 9, 6, 0, 7, 4, 7, 6, 5
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 10 2018

Keywords

Examples

			3.9561842030261697545408021818783008332999988095...

Crossrefs

Cf. A302598, A302700, A317350, A317351, A317355, A317356, A322737, A323311, A323313.

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

A317662 G.f.: Sum_{n>=0} ( (1+x)^n - 1 )^n * 2^n / (3 - 2*(1+x)^n)^(n+1).

Original entry on oeis.org

1, 2, 24, 448, 11820, 401392, 16668960, 818355488, 46367354632, 2977828665832, 213763450387456, 16961461169786752, 1474091484740240064, 139256465915227044352, 14208358055857371300864, 1557104405499802200814464, 182416569911688799401148816, 22749429746475540390909166048, 3009102958766163591152586574464, 420767787785916464100556297780608
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2018

Keywords

Comments

The following identities hold for |y| <= 1 and fixed real k > 0:
(C1) Sum_{n>=0} (y^n + k)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).
(C2) Sum_{n>=0} (y^n + 1)^n*k^n/(1+k + k*y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n*k^n/(1+k - k*y^n)^(n+1).
This sequence is an example of (C2) when y = 1+x and k = 2.

Examples

			G.f.: A(x) = 1 + 2*x + 24*x^2 + 448*x^3 + 11820*x^4 + 401392*x^5 + 16668960*x^6 + 818355488*x^7 + 46367354632*x^8 + ...
such that
A(x) = 1  +  ((1+x) - 1)*2/(3 - 2*(1+x))^2  +  ((1+x)^2 - 1)^2*2^2/(3 - 2*(1+x)^2)^3  +  ((1+x)^3 - 1)^3*2^3/(3 - 2*(1+x)^3)^4  +  ((1+x)^4 - 1)^4*2^4/(3 - 2*(1+x)^4)^5  +  ((1+x)^5 - 1)^5*2^5/(3 - 2*(1+x)^5)^6  +  ((1+x)^6 - 1)^6*2^6/(3 - 2*(1+x)^6)^7 + ...
Also,
A(x) = 1/5  +  ((1+x) + 1)*2/(3 + 2*(1+x))^2  +  ((1+x)^2 + 1)^2*2^2/(3 + 2*(1+x)^2)^3  +  ((1+x)^3 + 1)^3*2^3/(3 + 2*(1+x)^3)^4  +  ((1+x)^4 + 1)^4*2^4/(3 + 2*(1+x)^4)^5  +  ((1+x)^5 + 1)^5*2^5/(3 + 2*(1+x)^5)^6  +  ((1+x)^6 + 1)^6*2^6/(3 + 2*(1+x)^6)^7 + ...
EXAMPLE OF SUMS.
Evaluating the g.f. formally at x = -1/2, we obtain the sums
S1 = Sum_{n>=0} (1 - 2^n)^n * 4^n / (3*2^n - 2)^(n+1),
S2 = Sum_{n>=0} (1 + 2^n)^n * 4^n / (3*2^n + 2)^(n+1),
explicitly,
S1 = 1 - 4/4^2 + 3^2*4^2/10^3 - 7^3*4^3/22^4 + 15^4*4^4/46^5 - 31^5*4^5/94^6 + 63^6*4^6/190^7 - 127^7*4^7/382^8 + 255^8*4^8/766^9 - 511^9*4^9/1534^10 + 1023^10*4^10/3070^11 + ...
S2 = 1/5 + 3*4/8^2 + 5^2*4^2/14^3 + 9^3*4^3/26^4 + 17^4*4^4/50^5 + 33^5*4^5/98^6 + 65^6*4^6/194^7 + 129^7*4^7/386^8 + 257^8*4^8/770^9 + 513^9*4^9/1538^10 + 1025^10*4^10/3074^11 + ...
where S1 = S2 = 0.8378452129227094466992700455568437913726753230322...

Links

Crossrefs

Cf. A302598, A317663, A317664, A302614, A317350.

Programs

  • PARI
    {a(n) = my(A=1); A = sum(m=0, n, ( (1+x)^m - 1 +x*O(x^n) )^m * 2^m / (3 - 2*(1+x)^m +x*O(x^n) )^(m+1) ); ;polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( (1+x)^n - 1 )^n * 2^n / (3 - 2*(1+x)^n)^(n+1).
(2) A(x) = Sum_{n>=0} ( (1+x)^n + 1 )^n * 2^n / (3 + 2*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 7.5592435681748721825440151469382350654183499600538671407998439255608144356... and c = 0.30852178850187571906358489049387403704035769403106379389644818349... - Vaclav Kotesovec, Aug 09 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

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

A386665 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, 8, 90, 1336, 24406, 530234, 13410942, 388841734, 12762735148, 469004980720, 19105730068460, 855146084504046, 41724450644602328, 2204075802189470532, 125300401263988607716, 7626356269363721248332, 494723229572772238087966, 34070289390944902842701094, 2482276670026891882801017812
Offset: 0

Views

Author

Paul D. Hanna, Aug 29 2025

Keywords

Comments

It appears that lim_{n->oo} ( a(n+1)/a(n) )/(n+1) exists and is near 4.

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 90*x^3 + 1336*x^4 + 24406*x^5 + 530234*x^6 + 13410942*x^7 + 388841734*x^8 + 12762735148*x^9 + 469004980720*x^10 + ...
RELATED SERIES.
A(x)^(1/2) = 1 + 2*(x/4) + 62*(x/4)^2 + 2756*(x/4)^3 + 163574*(x/4)^4 + 11997852*(x/4)^5 + 1047984172*(x/4)^6 + 106571791752*(x/4)^7 + 12417003030694*(x/4)^8 + ...
A(x)^(-1/2) = 1 - 2*(x/4) - 58*(x/4)^2 - 2516*(x/4)^3 - 149434*(x/4)^4 - 11055996*(x/4)^5 - 976190180*(x/4)^6 - 100318703592*(x/4)^7 - 11796814729146*(x/4)^8 - ...

Links

Crossrefs

Cf. A322735, 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) = Sum_{n>=0} a(n)*x^n along with B(x) = A(x)^(1/2) satisfies the following formulas.
(1) A(x) = Sum_{n>=0} ( (1+x)^n - 1/B(x) )^n / ( 2 - (1+x)^n/B(x) )^(n+1).
(2) A(x) = Sum_{n>=0} ( (1+x)^n + 1/B(x) )^n / ( 2 + (1+x)^n/B(x) )^(n+1).
(3) B(x) = Sum_{n>=0} ( (1+x)^n*B(x) - 1 )^n / ( 2*B(x) - (1+x)^n )^(n+1).
(4) B(x) = Sum_{n>=0} ( (1+x)^n*B(x) + 1 )^n / ( 2*B(x) + (1+x)^n )^(n+1).

A322736 G.f. satisfies: A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - A(x)^(1/2) )^n / ( 3 - 2*(1+x)^n * A(x)^(1/2) )^(n+1).

Original entry on oeis.org

1, 2, 8, 96, 2956, 114992, 5244896, 277303392, 16680895688, 1124043943848, 83860544099264, 6863636560150656, 611673708807594944, 58982083391411043456, 6120766911879901270784, 680339106407429897733760, 80661483112436517009089168, 10162784535291704640507410016, 1356175692780348173552997926272, 191103836643650458447321745220736, 28358934286111202643351952170366400, 4420810085328675478052952299755080000
Offset: 0

Views

Author

Paul D. Hanna, Jan 25 2019

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 96*x^3 + 2956*x^4 + 114992*x^5 + 5244896*x^6 + 277303392*x^7 + 16680895688*x^8 + 1124043943848*x^9 + ...
such that A(x) and B = A(x)^(1/2) satisfy
A(x) = 1/(3 - 2*B)  +  2*((1+x) - B)/(3 - 2*(1+x)*B)^2  +  2^2*((1+x)^2 - B)^2/(3 - 2*(1+x)^2*B)^3  +  2^3*((1+x)^3 - B)^3/(3 - 2*(1+x)^3*B)^4  +  2^4*((1+x)^4 - B)^4/(3 - 2*(1+x)^4*B)^5  +  2^5*((1+x)^5 - B)^5/(3 - 2*(1+x)^5*B)^6 + ...
also,
A(x) = 1/(3 + 2*B)  +  2*((1+x) + B)/(3 + 2*(1+x)*B)^2  +  2^2*((1+x)^2 + B)^2/(3 + 2*(1+x)^2*B)^3  +  2^3*((1+x)^3 + B)^3/(3 + 2*(1+x)^3*B)^4  +  2^4*((1+x)^4 + B)^4/(3 + 2*(1+x)^4*B)^5  +  2^5*((1+x)^5 + B)^5/(3 + 2*(1+x)^5*B)^6 + ...

Crossrefs

Cf. A317350, A322735.

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)^(1/2) )^m *2^m / (3 - 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) and B(x) = A(x)^(1/2) satisfy:
(1) A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - B(x) )^n / ( 3 - 2*(1+x)^n * B(x) )^(n+1),
(2) A(x) = Sum_{n>=0} 2^n * ( (1+x)^n + B(x) )^n / ( 3 + 2*(1+x)^n * B(x) )^(n+1).
Showing 1-8 of 8 results.

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