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.

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

A303923 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^n - A(x) )^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 22, 92, 419, 2066, 10863, 60459, 354381, 2177439, 13979759, 93527819, 650509643, 4694372980, 35086564926, 271174745565, 2164066408692, 17808271012127, 150925549288155, 1315804758238582, 11787981398487995, 108409978503340041, 1022519935940220983, 9882436548778410911, 97788364370359938816
Offset: 0

Views

Author

Paul D. Hanna, May 03 2018

Keywords

Comments

Compare to: 1 = Sum_{n>=0} ( 1 + x*G(x)^k - G(x) )^n holds trivially for fixed k>0 when G(x) = 1 + x*G(x)^k ; this sequence explores the case when k varies with n.

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 22*x^6 + 92*x^7 + 419*x^8 + 2066*x^9 + 10863*x^10 + 60459*x^11 + 354381*x^12 + ...
such that
1 = 1 + (1 + x*A(x) - A(x)) + (1 + x*A(x)^2 - A(x))^2 + (1 + x*A(x)^3 - A(x))^3 + (1 + x*A(x)^4 - A(x))^4 + (1 + x*A(x)^5 - A(x))^5 + ...

Links

Crossrefs

Cf. A303924, A303925, A303926.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( 1 + x*Ser(A)^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*A(x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} x^n * A(x)^(n^2) / (1 + (A(x)-1)*A(x)^n)^(n+1). - Paul D. Hanna, Dec 11 2018
G.f.: x/Series_Reversion( x*F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^(n+1) - F(x))^n, where F(x) is the g.f. of A303924.
G.f.: sqrt( x/Series_Reversion( x*G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^(n+2) - G(x))^n, where G(x) is the g.f. of A303925.

A303927 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x)^2)^n - A(x) )^n.

Original entry on oeis.org

1, 1, 3, 19, 199, 2863, 51280, 1087107, 26492959, 728234294, 22273547313, 750180870861, 27591387247199, 1100527782602563, 47324815446060104, 2182852921566858499, 107515416285928793865, 5632697086212688424650, 312779421789041421062682, 18351511395587408908636348, 1134459736825581425674735933
Offset: 0

Views

Author

Paul D. Hanna, May 03 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 199*x^4 + 2863*x^5 + 51280*x^6 + 1087107*x^7 + 26492959*x^8 + 728234294*x^9 + 22273547313*x^10 + ...
such that
1 = 1 + ((1 + x*A(x)^2) - A(x)) + ((1 + x*A(x)^2)^2 - A(x))^2 + ((1 + x*A(x)^2)^3 - A(x))^3 + ((1 + x*A(x)^2)^4 - A(x))^4 + ((1 + x*A(x)^2)^5 - A(x))^5 + ...

Crossrefs

Cf. A303926, A303928.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( (1 + x*Ser(A)^2)^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*A(x)^2)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} (1 + x*A(x)^2)^(n^2) / (1 + A(x)*(1 + x*A(x)^2)^n)^(n+1). - Paul D. Hanna, Dec 11 2018
G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^n - F(x))^n, where F(x) is the g.f. of A303926.
G.f.: x/Series_Reversion( x*G(x) ) such that 1 = Sum_{n>=0} ((1 + x*G(x)^3)^n - G(x))^n, where G(x) is the g.f. of A303928.

A303928 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x)^3)^n - A(x) )^n.

Original entry on oeis.org

1, 1, 4, 29, 312, 4454, 78649, 1644280, 39580036, 1076460972, 32628557331, 1090654903233, 39861817143230, 1581648436369772, 67718096677762406, 3112120229328860775, 152815413664021339930, 7985028281346030147672, 442406826626726978612624, 25906474516335623637923581, 1598761621228278791567817906
Offset: 0

Views

Author

Paul D. Hanna, May 03 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 312*x^4 + 4454*x^5 + 78649*x^6 + 1644280*x^7 + 39580036*x^8 + 1076460972*x^9 + 32628557331*x^10 + ...
such that
1 = 1 + ((1 + x*A(x)^3) - A(x)) + ((1 + x*A(x)^3)^2 - A(x))^2 + ((1 + x*A(x)^3)^3 - A(x))^3 + ((1 + x*A(x)^3)^4 - A(x))^4 + ((1 + x*A(x)^3)^5 - A(x))^5 + ...

Crossrefs

Cf. A303926, A303927.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( (1 + x*Ser(A)^3)^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*A(x)^3)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} (1 + x*A(x)^3)^(n^2) / (1 + A(x)*(1 + x*A(x)^3)^n)^(n+1). - Paul D. Hanna, Dec 11 2018
G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x)^2)^n - F(x))^n, where F(x) is the g.f. of A303927.
G.f.: sqrt( 1/x*Series_Reversion( x/G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^n - G(x))^n, where G(x) is the g.f. of A303926.
Showing 1-4 of 4 results.

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