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 1-3 of 3 results.

A303056 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 1, 8, 89, 1326, 24247, 521764, 12867985, 357229785, 11017306489, 373675921093, 13825260663882, 554216064798423, 23934356706763264, 1108017262467214486, 54747529760516714323, 2876096694574711401525, 160092696678371426933342, 9413031424290635395882462, 583000844360279565483710624
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2018

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 1, p = -A(x), q = (1+x). - Paul D. Hanna, Jun 22 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 89*x^4 + 1326*x^5 + 24247*x^6 + 521764*x^7 + 12867985*x^8 + 357229785*x^9 + 11017306489*x^10 + ...
such that
1 = 1  +  ((1+x) - A(x))  +  ((1+x)^2 - A(x))^2  +  ((1+x)^3 - A(x))^3  +  ((1+x)^4 - A(x))^4  +  ((1+x)^5 - A(x))^5  +  ((1+x)^6 - A(x))^6  +  ((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)/(1 + (1+x)*A(x))^2  +  (1+x)^4/(1 + (1+x)^2*A(x))^3  +  (1+x)^9/(1 + (1+x)^3*A(x))^4  +  (1+x)^16/(1 + (1+x)^4*A(x))^5  +  (1+x)^25/(1 + (1+x)^5*A(x))^6  +  (1+x)^36/(1 + (1+x)^6*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = x + x^2/2 + 22*x^3/3 + 325*x^4/4 + 6186*x^5/5 + 137380*x^6/6 + 3478651*x^7/7 + 98674253*x^8/8 + 3096911434*x^9/9 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1/2, it may be evaluated formally.
Let t = A(-1/2) = 0.545218973635949431234950245034944106957612798888179456724264...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).
Also,
A(r) = 1/2 at r = -0.54683649902292991492196620520872286547799291909992048564578...
where
(1) 1 = Sum_{n>=0} ( (1+r)^n - 1/2 )^n.
(2) 1 = Sum_{n>=0} (1+r)^(-n) / ( 1/(1+r)^n + 1/2 )^(n+1).

Links

Crossrefs

Cf. A304642, A304639, A303926.
Cf. A321602, A321603, A321604, A321605.
Cf. A326282, A326283, A326284.
Cf. A337755, A337756, A337757.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ((1+x)^m - Ser(A))^m ) )[#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 - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654... and c = 0.11739505492506... - Vaclav Kotesovec, Sep 26 2020

A326282 G.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 2, 28, 586, 16336, 559164, 22519620, 1039209116, 53968031108, 3112841732920, 197413519635632, 13654508980460736, 1023144120035225664, 82581014079320743504, 7144332294806845079568, 659630258631919908187784, 64748755209330058463666656, 6733915902264715745675338784, 739732094650896407811045989408, 85594689069528757090534336595600
Offset: 0

Views

Author

Paul D. Hanna, Jun 22 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 2, p = -A(x), q = (1+x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 28*x^3 + 586*x^4 + 16336*x^5 + 559164*x^6 + 22519620*x^7 + 1039209116*x^8 + 53968031108*x^9 + 3112841732920*x^10 + ...
such that
1 = 1  +  2*((1+x) - A(x))  +  2^2*((1+x)^2 - A(x))^2  +  2^3*((1+x)^3 - A(x))^3  +  2^4*((1+x)^4 - A(x))^4  +  2^5*((1+x)^5 - A(x))^5  +  2^6*((1+x)^6 - A(x))^6  +  2^7*((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + 2*A(x))  +  2*(1+x)/(1 + 2*(1+x)*A(x))^2  +  2^2*(1+x)^4/(1 + 2*(1+x)^2*A(x))^3  +  2^3*(1+x)^9/(1 + 2*(1+x)^3*A(x))^4  +  2^4*(1+x)^16/(1 + 2*(1+x)^4*A(x))^5  +  2^5*(1+x)^25/(1 + 2*(1+x)^5*A(x))^6  +  2^6*(1+x)^36/(1 + 2*(1+x)^6*A(x))^7 + ...

Links

Crossrefs

Cf. A303056, A326283, A326284.

Programs

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

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} 2^n * ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} 2^n * (1+x)^(n^2) / (1 + 2*(1+x)^n*A(x))^(n+1).
a(n) ~ c * (1 + 2*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r = 0.925556278640887084941460444526398190071550948416... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/2 and c = 0.06492129653731... - Vaclav Kotesovec, Oct 13 2020

A326284 G.f. A(x) satisfies: 1 = Sum_{n>=0} 4^n * ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 4, 104, 4196, 225216, 14845072, 1151255440, 102289538128, 10226417550096, 1135388485042624, 138583671424928128, 18446474604149746176, 2659732597343823233280, 413060592233577210697984, 68754628660531280009195776, 12213125156726936259944672320, 2306358043375070604869802287616, 461443265563759624969778550969344, 97514484569091438266511351355560448
Offset: 0

Views

Author

Paul D. Hanna, Jun 22 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 4, p = -A(x), q = (1+x).

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 104*x^3 + 4196*x^4 + 225216*x^5 + 14845072*x^6 + 1151255440*x^7 + 102289538128*x^8 + 10226417550096*x^9 + 1135388485042624*x^10 + ...
such that
1 = 1  +  4*((1+x) - A(x))  +  4^2*((1+x)^2 - A(x))^2  +  4^3*((1+x)^3 - A(x))^3  +  4^4*((1+x)^4 - A(x))^4  +  4^5*((1+x)^5 - A(x))^5  +  4^6*((1+x)^6 - A(x))^6  +  4^7*((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + 4*A(x))  +  4*(1+x)/(1 + 4*(1+x)*A(x))^2  +  4^2*(1+x)^4/(1 + 4*(1+x)^2*A(x))^3  +  4^3*(1+x)^9/(1 + 4*(1+x)^3*A(x))^4  +  4^4*(1+x)^16/(1 + 4*(1+x)^4*A(x))^5  +  4^5*(1+x)^25/(1 + 4*(1+x)^5*A(x))^6  +  4^6*(1+x)^36/(1 + 4*(1+x)^6*A(x))^7 + ...

Links

Crossrefs

Cf. A303056, A326282, A326283.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 4^m*((1+x)^m - Ser(A))^m ) )[#A]/4 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} 4^n * ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} 4^n * (1+x)^(n^2) / (1 + 4*(1+x)^n*A(x))^(n+1).
a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.034391206985341... - Vaclav Kotesovec, Oct 13 2020
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.