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

A317340 G.f. A(x) satisfies: 1+x = Sum_{n>=0} ((1+x)^(n+1) - A(x))^n.

Original entry on oeis.org

1, 1, 5, 19, 215, 3221, 60255, 1328529, 33525557, 949932155, 29820028655, 1026798211345, 38478492170097, 1559272992752559, 67959179777259427, 3170787800382000429, 157723858411070076821, 8334021579932292731707, 466249346920348934320021, 27536331208734157654465035, 1712202839967112789793314087, 111816679758439520925160452089
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 19*x^3 + 215*x^4 + 3221*x^5 + 60255*x^6 + 1328529*x^7 + 33525557*x^8 + 949932155*x^9 + 29820028655*x^10 + ...
such that
1+x = 1  +  ((1+x)^2 - A(x))  +  ((1+x)^3 - A(x))^2  +  ((1+x)^4 - A(x))^3  +  ((1+x)^5 - A(x))^4  +  ((1+x)^6 - A(x))^5  +  ((1+x)^7 - A(x))^6  +  ((1+x)^8 - A(x))^7 + ...
Also,
1+x = 1/(1 + A(x))  +  (1+x)^2/(1 + (1+x)*A(x))^2  +  (1+x)^6/(1 + (1+x)^2*A(x))^3  +  (1+x)^12/(1 + (1+x)^3*A(x))^4  +  (1+x)^20/(1 + (1+x)^4*A(x))^5  +  (1+x)^30/(1 + (1+x)^5*A(x))^6  +  (1+x)^42/(1 + (1+x)^6*A(x))^7 + ...

Links

Crossrefs

Cf. A304642, A303056.

Programs

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

Formula

G.f. A(x) satisfies:
(1) 1+x = Sum_{n>=0} ((1+x)^(n+1) - A(x))^n.
(2) 1+x = Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * r^(2*n) * (1 + exp(1/r))^n * n^n / exp(n), where r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation (1 + exp(-1/r)) * LambertW(-exp(-1/r)/r) = -1/r, and c = 0.9243009841585606... - Vaclav Kotesovec, Aug 06 2018

A304641 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n.

Original entry on oeis.org

1, 2, 6, 74, 3078, 228842, 25277286, 3837501194, 762731347398, 191798593122602, 59475206565622566, 22290155840476400714, 9933314218291366691718, 5192540728710234994272362, 3147427468437058629798524646, 2190237887318737512524514442634, 1734606000858253287464231519860038, 1551466530739217915273113571521758122, 1556475858078242120174483544923467343526
Offset: 0

Views

Author

Paul D. Hanna, May 16 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 74*x^3/3! + 3078*x^4/4! + 228842*x^5/5! + 25277286*x^6/6! + 3837501194*x^7/7! + 762731347398*x^8/8! + 191798593122602*x^9/9! + 59475206565622566*x^10/10! + ...
such that
1 = 1  +  (exp(2*x) - A(x))  +  (exp(3*x) - A(x))^2  +  (exp(4*x) - A(x))^3  +  (exp(5*x) - A(x))^4  +  (exp(6*x) - A(x))^5  +  (exp(7*x) - A(x))^6  +  (exp(8*x) - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  exp(2*x)/(1 + exp(x)*A(x))^2  +  exp(6*x)/(1 + exp(2*x)*A(x))^3  +  exp(12*x)/(1 + exp(3*x)*A(x))^4  +  exp(20*x)/(1 + exp(4*x)*A(x))^5  +  exp(30*x)/(1 + exp(5*x)*A(x))^6  +  exp(42*x)/(1 + exp(6*x)*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = 2*x + 2*x^2/2! + 54*x^3/3! + 2570*x^4/4! + 199590*x^5/5! + 22598762*x^6/6! + 3488755494*x^7/7! + 701959131050*x^8/8! + 178186466260710*x^9/9! + 55669778154059882*x^10/10! + ...
exp(-x) * A(x) = 1 + x + 3*x^2/2! + 61*x^3/3! + 2811*x^4/4! + 214141*x^5/5! + 23949003*x^6/6! + 3665260621*x^7/7! + 732726498171*x^8/8! + 185070066199261*x^9/9! + 57591088296085803*x^10/10! + ...

Crossrefs

Cf. A304640, A304642.

Programs

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

Formula

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