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 1-5 of 5 results.

A326091 E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n!.

Original entry on oeis.org

1, 3, 11, 66, 601, 7418, 116505, 2248522, 52025473, 1414524690, 44471074249, 1595792690594, 64659403375137, 2931455146804330, 147550017664392457, 8189594420467104042, 498288959815836863233, 33061714451161940667554, 2381086262720126177230473, 185362512554618232339122578, 15539467373234774634135507361, 1398111233425766921500901239098, 134584560980879138160145116701257
Offset: 0

Views

Author

Paul D. Hanna, Jun 28 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x) with p = 2, r = x.

Examples

			E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 66*x^3/3! + 601*x^4/4! + 7418*x^5/5! + 116505*x^6/6! + 2248522*x^7/7! + 52025473*x^8/8! + 1414524690*x^9/9! + 44471074249*x^10/10! + ...
such that
A(x) = 1 + (2 + exp(x))*x + (2 + exp(2*x))^2*x^2/2! + (2 + exp(3*x))^3*x^3/3! + (2 + exp(4*x))^4*x^4/4! + (2 + exp(5*x))^5*x^5/5! + (2 + exp(6*x))^6*x^6/6! + ...
also
A(x) = exp(2*x) + exp(x + 2*exp(x)*x)*x + exp(4*x + 2*exp(2*x)*x)*x^2/2! + exp(9*x + 2*exp(3*x)*x)*x^3/3! + exp(16*x + 2*exp(4*x)*x)*x^4/4! + exp(25*x + 2*exp(5*x)*x)*x^5/5! + exp(36*x + 2*exp(6*x)*x)*x^6/6! + ...

Links

Crossrefs

Cf. A108459, A326090, A326261.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (2 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} exp( n^2*x + 2*exp(n*x)*x ) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m^2*x + 2*exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

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

A326261 E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n!.

Original entry on oeis.org

1, 4, 18, 115, 1076, 13749, 223342, 4437115, 105308472, 2930229721, 94110395546, 3444510650343, 142161931150564, 6557368148307253, 335460464343013494, 18907437932151629899, 1167279375125285092592, 78529603970775837111729, 5730854443905658384812466, 451803953552256670477653679, 38337003901469883140928003036, 3489532046271886600931347767373
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x) with p = 3, r = x.

Examples

			E.g.f.: A(x) = 1 + 4*x + 18*x^2/2! + 115*x^3/3! + 1076*x^4/4! + 13749*x^5/5! + 223342*x^6/6! + 4437115*x^7/7! + 105308472*x^8/8! + 2930229721*x^9/9! + 94110395546*x^10/10! + ...
such that
A(x) = 1 + (3 + exp(x))*x + (3 + exp(2*x))^2*x^2/2! + (3 + exp(3*x))^3*x^3/3! + (3 + exp(4*x))^4*x^4/4! + (3 + exp(5*x))^5*x^5/5! + (3 + exp(6*x))^6*x^6/6! + ...
also
A(x) = exp(3*x) + exp(x + 3*exp(x)*x)*x + exp(4*x + 3*exp(2*x)*x)*x^2/2! + exp(9*x + 3*exp(3*x)*x)*x^3/3! + exp(16*x + 3*exp(4*x)*x)*x^4/4! + exp(25*x + 3*exp(5*x)*x)*x^5/5! + exp(36*x + 3*exp(6*x)*x)*x^6/6! + ...

Links

Crossrefs

Cf. A108459, A326090, A326091.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (3 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} exp( n^2*x + 3*exp(n*x)*x ) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m^2*x + 3*exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n!.
E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( 3*exp(n*x)*x ) * x^n/n!.

A326009 E.g.f.: Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!.

Original entry on oeis.org

1, 2, 8, 56, 564, 7452, 124126, 2527646, 61337576, 1740438008, 56893173354, 2116141180650, 88637462278492, 4144712080864292, 214742915441526686, 12247719772739219558, 764573919234220965072, 51977513845734053953776, 3830761480589037404767954, 304839727443701572462549058, 26096983659506717348854764356, 2395544800795092178844224643612, 235073598248121646307555752669446
Offset: 0

Views

Author

Paul D. Hanna, Jul 13 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x), p = exp(-x), r = exp(x)*x.

Examples

			E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 564*x^4/4! + 7452*x^5/5! + 124126*x^6/6! + 2527646*x^7/7! + 61337576*x^8/8! + 1740438008*x^9/9! + 56893173354*x^10/10! + ...
such that
A(x) = 1 + (exp(2*x) + 1)*x + (exp(3*x) + 1)^2*x^2/2! + (exp(4*x) + 1)^3*x^3/3! + (exp(5*x) + 1)^4*x^4/4! + (exp(6*x) + 1)^5*x^5/5! + ...
also
A(x) = exp(x) + exp(2*x)*exp(exp(x)*x)*x + exp(6*x)*exp(exp(2*x)*x)*x^2/2! + exp(12*x)*exp(exp(3*x)*x)*x^3/3! + exp(20*x)*exp(exp(4*x)*x)*x^4/4! + ...

Crossrefs

Cf. A326090, A326550.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (1 + exp((n+1)*x))^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (1 + exp((m+1)*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} exp(n*(n+1)*x) * exp(exp(n*x)*x) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m*(m+1)*x + exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! equals the following sums.
(1) Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!,
(2) Sum_{n>=0} exp(n*(n+1)*x) * exp(exp(n*x)*x) * x^n / n!.

A356772 E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} ( x^n + x*A(x) )^n / n!.

Original entry on oeis.org

1, 2, 5, 34, 329, 3716, 55777, 1010206, 21187049, 511352272, 13929248861, 422450642054, 14129873671069, 516664310959720, 20503766568423881, 877759284120870526, 40321132468408643153, 1978363648482263649728, 103262474042895179595061, 5713315282015940379009862
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2022

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = x with p = x*A(x), r = 1.

Examples

			E.g.f.: A(x) = 1 + 2*x + 5*x^2/2! + 34*x^3/3! + 329*x^4/4! + 3716*x^5/5! + 55777*x^6/6! + 1010206*x^7/7! + 21187049*x^8/8! + 511352272*x^9/9! + 13929248861*x^10/10! + ...
where
A(x) = 1 + (x + x*A(x)) + (x^2 + x*A(x))^2/2! + (x^3 + x*A(x))^3/3! + (x^4 + x*A(x))^4/4! + (x^5 + x*A(x))^5/5! + ... + (x^n + x*A(x))^n/n! + ...
also
A(x) = exp(x*A(x)) + x*exp(x^2*A(x)) + x^4*exp(x^3*A(x))/2! + x^9*exp(x^4*A(x))/3! + x^16*exp(x^5*A(x))/4! + x^25*exp(x^6*A(x))/5! + ... +  + x^(n^2)*exp(x^(n+1)*A(x))/n! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 5*x^2/2! + 28*x^3/3! + 269*x^4/4! + 3356*x^5/5! + 50257*x^6/6! + 915076*x^7/7! + 19427753*x^8/8! + 471310984*x^9/9! + 12892968701*x^10/10! + ...
log(A(x)) = 2*x + x^2/2! + 20*x^3/3! + 126*x^4/4! + 1314*x^5/5! + 20460*x^6/6! + 347906*x^7/7! + 7181944*x^8/8! + 170606106*x^9/9! + 4577504760*x^10/10! + ...
SPECIFIC VALUES.
A(x = 1/4) = 1.8854569251645435475372616427080...
A(x = 0.3) = 2.4910587821818158559566392662113...
A(x = 1/3) diverges.

Links

Crossrefs

Cf. A356773, A108459, A326090, A326091, A326261, A326009.

Programs

  • PARI
    /* A(x) = Sum_{n>=0} ( x^n + x*A(x) )^n / n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, (x^m + x*A +x*O(x^n))^m/m! )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* A(x) = Sum_{n>=0} x^(n^2) * exp(x^(n+1)*A(x))/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,sqrtint(n), x^(m^2) * exp( x^(m+1)*A +x*O(x^n)) / m! )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = Sum_{n>=0} ( x^n + x*A(x) )^n / n!.
(2) A(x) = Sum_{n>=0} x^(n^2) * exp( x^(n+1) * A(x) ) / n!.
a(n) ~ c * d^n * n! / n^(3/2), where d = 3.14614463757985697... and c = 1.454175198420213... - Vaclav Kotesovec, Jul 03 2025

A356773 E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} ( x^n + A(x) )^n * x^n / n!.

Original entry on oeis.org

1, 1, 5, 22, 197, 2076, 29527, 477394, 9248745, 204340600, 5111234891, 142148945214, 4362830874877, 146338813894612, 5328688224075231, 209295914833477546, 8821420994034588113, 397128156446044087536, 19019218255697847951955, 965527468715744517674998
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2022

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = x with p = A(x), r = x.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 22*x^3/3! + 197*x^4/4! + 2076*x^5/5! + 29527*x^6/6! + 477394*x^7/7! + 9248745*x^8/8! + 204340600*x^9/9! + 5111234891*x^10/10! + ...
where
A(x) = 1 + (x + A(x))*x + (x^2 + A(x))^2*x^2/2! + (x^3 + A(x))^3*x^3/3! + (x^4 + A(x))^4*x^4/4! + (x^5 + A(x))^5*x^5/5! + ... + (x^n + A(x))^n*x^n/n! + ...
also
A(x) = exp(x*A(x)) + x^2*exp(x^2*A(x)) + x^6*exp(x^3*A(x))/2! + x^12*exp(x^4*A(x))/3! + x^20*exp(x^5*A(x))/4! + x^30*exp(x^6*A(x))/5! + ... +  x^(n*(n+1))*exp(x^(n+1)*A(x))/n! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 173*x^4/4! + 1956*x^5/5! + 27007*x^6/6! + 453874*x^7/7! + 8790105*x^8/8! + 195462136*x^9/9! + 4899670811*x^10/10! + ...
log(A(x)) = x + 4*x^2/2! + 9*x^3/3! + 88*x^4/4! + 905*x^5/5! + 12606*x^6/6! + 189217*x^7/7! + 3600472*x^8/8! + 78839217*x^9/9! + 1944056890*x^10/10! + ...
SPECIFIC VALUES.
A(x = 1/4) = 1.5376989442827462484156603674393740195...
A(x = 1/3) = 2.2880218830072453104841119982317247920...
A(x = 0.4) diverges.

Links

Crossrefs

Cf. A356772, A108459, A326090, A326091, A326261, A326009.

Programs

  • PARI
    /* A(x) = Sum_{n>=0} ( x^n + A(x) )^n * x^n / n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, (x^m + A +x*O(x^n))^m*x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* A(x) = Sum_{n>=0} x^(n*(n+1)) * exp(x^(n+1)*A(x))/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,sqrtint(n), x^(m*(m+1)) * exp( x^(m+1)*A +x*O(x^n)) / m! )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = Sum_{n>=0} ( x^n + A(x) )^n * x^n / n!.
(2) A(x) = Sum_{n>=0} x^(n*(n+1)) * exp( x^(n+1) * A(x) ) / n!.
a(n) ~ c * d^n * n! / n^(3/2), where d = 2.88676786838244269... and c = 1.26061634709684... - Vaclav Kotesovec, Jul 03 2025
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