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.

Previous Showing 11-17 of 17 results.

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

A337755 G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1) * 2^n * ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 3, 52, 1320, 43440, 1722712, 79186272, 4118457732, 238450436416, 15189543467688, 1055122226778720, 79359687454230296, 6425540170275120528, 557306222539540276176, 51558846502494563714080, 5068865533417385007925076, 527798429103621760357553448, 58032815995114574483132149504
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2020

Keywords

Comments

In general, 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 = 2 with r = 2, p = -A(x), q = (1+x).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 52*x^3 + 1320*x^4 + 43440*x^5 + 1722712*x^6 + 79186272*x^7 + 4118457732*x^8 + 238450436416*x^9 + ...
where
1 = 1  +  2*2*((1+x) - A(x))  +  3*2^2*((1+x)^2 - A(x))^2  +  4*2^3*((1+x)^3 - A(x))^3  +  5*2^4*((1+x)^4 - A(x))^4  +  6*2^5*((1+x)^5 - A(x))^5  +  7*2^6*((1+x)^6 - A(x))^6  +  8*2^7*((1+x)^7 - A(x))^7 + ... + (n+1)*2^n*((1+x)^n - A(x))^n + ...
Also,
1 = 1/(1 + 2*A(x))^2  +  2*2*(1+x)/(1 + 2*(1+x)*A(x))^3  +  3*2^2*(1+x)^4/(1 + 2*(1+x)^2*A(x))^4  +  4*2^3*(1+x)^9/(1 + 2*(1+x)^3*A(x))^5  +  5*2^4*(1+x)^16/(1 + 2*(1+x)^4*A(x))^6  +  6*2^5*(1+x)^25/(1 + 2*(1+x)^5*A(x))^7  +  7*2^6*(1+x)^36/(1 + 2*(1+x)^6*A(x))^8 + ... + (n+1)*2^n*(1+x)^(n^2)/(1 + 2*(1+x)^n*A(x))^(n+2) + ...

Links

Crossrefs

Cf. A303056, A337756, A337757, A301581.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (m+1) * 2^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} (n+1) * 2^n * ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} (n+1) * 2^n * (1+x)^(n^2) / (1 + 2*(1+x)^n*A(x))^(n+2).
a(n) ~ c * d^n * n! * sqrt(n), where d = (1 + 2*exp(1/r)) * r^2 = 5.9039765456700218004884947864345974878995107118996038057..., 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 = 0.0300441568138... - Vaclav Kotesovec, Sep 25 2020

A337756 G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)*(n+2)/2! * 3^n * ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 6, 180, 7845, 434448, 28594494, 2157238350, 182404049175, 17026549342770, 1735705779016158, 191667825521201286, 22781050822806698709, 2899308092950790588988, 393385952195184523370994, 56691647586489579559334352, 8649001755741912766806347253, 1392791055204268736953260163092
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2020

Keywords

Comments

In general, 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 = 3 with r = 3, p = -A(x), q = (1+x).

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 180*x^3 + 7845*x^4 + 434448*x^5 + 28594494*x^6 + 2157238350*x^7 + 182404049175*x^8 + ...
where
1 = 1  +  3*3*((1+x) - A(x))  +  6*3^2*((1+x)^2 - A(x))^2  +  10*3^3*((1+x)^3 - A(x))^3  +  15*3^4*((1+x)^4 - A(x))^4  +  21*3^5*((1+x)^5 - A(x))^5  +  28*3^6*((1+x)^6 - A(x))^6  +  38*3^7*((1+x)^7 - A(x))^7 + ... + C(n+2,2)*3^n*((1+x)^n - A(x))^n + ...
Also,
1 = 1/(1 + 3*A(x))^3  +  3*3*(1+x)/(1 + 3*(1+x)*A(x))^4  +  6*3^2*(1+x)^4/(1 + 3*(1+x)^2*A(x))^5  +  10*3^3*(1+x)^9/(1 + 3*(1+x)^3*A(x))^6  +  15*3^4*(1+x)^16/(1 + 3*(1+x)^4*A(x))^7  +  21*3^5*(1+x)^25/(1 + 3*(1+x)^5*A(x))^8  +  28*3^6*(1+x)^36/(1 + 3*(1+x)^6*A(x))^9 + ... + C(n+2,2)*3^n*(1+x)^(n^2)/(1 + 3*(1+x)^n*A(x))^(n+3) + ...

Links

Crossrefs

Cf. A303056, A337755, A337757.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (m+1)*(m+2)/2! * 3^m * ((1+x)^m - Ser(A))^m ) )[#A]/9 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} C(n+2,2) * 3^n * ( (1+x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} C(n+2,2) * 3^n * (1+x)^(n^2) / (1 + 3*(1+x)^n*A(x))^(n+3).
a(n) ~ c * (1 + 3*exp(1/r))^n * r^(2*n) * n! * n^(3/2), where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.00671991787239... - Vaclav Kotesovec, Oct 13 2020

A337757 G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * 4^n * ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 10, 460, 30250, 2488776, 240707480, 26452491760, 3233941091480, 433611348176880, 63118887464611936, 9899442124162104960, 1662993951689377716800, 297806177944353392091200, 56626969607275080551099520, 11394470658417110387020266496, 2419172929237326590857901776560, 540511078482106447677809541679680
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2020

Keywords

Comments

In general, 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 = 4 with r = 4, p = -A(x), q = (1+x).

Examples

			G.f.: A(x) = 1 + x + 10*x^2 + 460*x^3 + 30250*x^4 + 2488776*x^5 + 240707480*x^6 + 26452491760*x^7 + 3233941091480*x^8 + ...
where
1 = 1  +  4*4*((1+x) - A(x))  +  10*4^2*((1+x)^2 - A(x))^2  +  20*4^3*((1+x)^3 - A(x))^3  +  35*4^4*((1+x)^4 - A(x))^4  +  56*4^5*((1+x)^5 - A(x))^5  +  84*4^6*((1+x)^6 - A(x))^6  +  120*4^7*((1+x)^7 - A(x))^7 + ... + C(n+3,3)*4^n*((1+x)^n - A(x))^n + ...
Also,
1 = 1/(1 + 4*A(x))^4  +  4*4*(1+x)/(1 + 4*(1+x)*A(x))^5  +  10*4^2*(1+x)^4/(1 + (1+x)^2*A(x))^6  +  20*4^3*(1+x)^9/(1 + 4*(1+x)^3*A(x))^7  +  35*4^4*(1+x)^16/(1 + 4*(1+x)^4*A(x))^8  +  56*4^5*(1+x)^25/(1 + 4*(1+x)^5*A(x))^9  +  84*4^6*(1+x)^36/(1 + 4*(1+x)^6*A(x))^10 + ... + C(n+3,3)*4^n*(1+x)^(n^2)/(1 + 4*(1+x)^n*A(x))^(n+4) + ...

Links

Crossrefs

Cf. A303056, A337755, A337756.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (m+1)*(m+2)*(m+3)/3! * 4^m * ((1+x)^m - Ser(A))^m ) )[#A]/16 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} C(n+3,3) * 4^n * ( (1+x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} C(n+3,3) * 4^n * (1+x)^(n^2) / (1 + 4*(1+x)^n*A(x))^(n+4).
a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n! * n^(5/2), 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 = 0.0012636042138... - Vaclav Kotesovec, Oct 13 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

A304640 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( exp(n*x) - A(x) )^n.

Original entry on oeis.org

1, 1, 3, 55, 2439, 181711, 19987863, 3019344175, 597283032279, 149571915236911, 46218017081300823, 17270884763586798895, 7677911426885078360919, 4005536546107407400763311, 2423921346754787141028928983, 1684444421472099056470715447215, 1332493495574767096115773084870359, 1190644894731926448479445174157508911, 1193491123893325068744832273320725408343
Offset: 0

Views

Author

Paul D. Hanna, May 16 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 55*x^3/3! + 2439*x^4/4! + 181711*x^5/5! + 19987863*x^6/6! + 3019344175*x^7/7! + 597283032279*x^8/8! + 149571915236911*x^9/9! + 46218017081300823*x^10/10! + ...
such that
1 = 1  +  (exp(x) - A(x))  +  (exp(2*x) - A(x))^2  +  (exp(3*x) - A(x))^3  +  (exp(4*x) - A(x))^4  +  (exp(5*x) - A(x))^5  +  (exp(6*x) - A(x))^6  +  (exp(7*x) - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  exp(x)/(1 + exp(x)*A(x))^2  +  exp(4*x)/(1 + exp(2*x)*A(x))^3  +  exp(9*x)/(1 + exp(3*x)*A(x))^4  +  exp(16*x)/(1 + exp(4*x)*A(x))^5  +  exp(25*x)/(1 + exp(5*x)*A(x))^6  +  exp(36*x)/(1 + exp(6*x)*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = x + 2*x^2/2! + 48*x^3/3! + 2222*x^4/4! + 169080*x^5/5! + 18843302*x^6/6! + 2872307088*x^7/7! + 571992255662*x^8/8! + 143972732107560*x^9/9! + 44668284142577462*x^10/10! + ...
The derivative of e.g.f. A(x) equals the ratio of the series:
A'(x) = [ Sum_{n>=1} n^2 * exp(n*x) * ( exp(n*x) - A(x) )^(n-1) ] / [ Sum_{n>=1} n * ( exp(n*x) - A(x) )^(n-1) ]. - _Paul D. Hanna_, Aug 06 2018

Crossrefs

Cf. A303056, A304641, A305134.

Programs

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

Formula

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

A325154 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(2*n-1) - A(x))^n.

Original entry on oeis.org

1, 1, 4, 60, 1349, 40210, 1470027, 63225750, 3116555468, 172936040306, 10661699020596, 722933543336296, 53476543241702021, 4286318739039468220, 370139507278333619231, 34264675353237245461705, 3385595826616475280589858, 355676742010175185149150523, 39592541401227701053287450374, 4655516336942715288212969823798, 576645913391345319618489456738288, 75048370900002385430200781452328814
Offset: 0

Views

Author

Paul D. Hanna, Apr 12 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 60*x^3 + 1349*x^4 + 40210*x^5 + 1470027*x^6 + 63225750*x^7 + 3116555468*x^8 + 172936040306*x^9 + 10661699020596*x^10 + ...
such that
1 = 1  +  ((1+x) - A(x))  +  ((1+x)^3 - A(x))^2  +  ((1+x)^5 - A(x))^3  +  ((1+x)^7 - A(x))^4  +  ((1+x)^9 - A(x))^5  +  ((1+x)^11 - A(x))^6  +  ((1+x)^13 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)/(1 + (1+x)^2*A(x))^2  +  (1+x)^6/(1 + (1+x)^4*A(x))^3  +  (1+x)^15/(1 + (1+x)^6*A(x))^4  +  (1+x)^28/(1 + (1+x)^8*A(x))^5  +  (1+x)^45/(1 + (1+x)^10*A(x))^6  +  (1+x)^66/(1 + (1+x)^6*A(x))^7 + ...

Crossrefs

Cf. A303056.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(2*m-1) - 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)^(2*n-1) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(2*n^2-n) / (1 + (1+x)^(2*n)*A(x))^(n+1).
Previous Showing 11-17 of 17 results.

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