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.

A326090 E.g.f.: Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!.

Original entry on oeis.org

1, 2, 6, 35, 308, 3637, 55150, 1033027, 23260536, 617066297, 18968614874, 666664879663, 26496140541700, 1179815542970053, 58388906382906390, 3189604848766578563, 191168734534622234480, 12504288586619417431921, 888401197086798248554546, 68270033412187747029025111, 5652853046029263008213465916, 502601914954325406783531231677, 47834047958592244085651443711406
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 = 1, r = x.

Examples

			E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 35*x^3/3! + 308*x^4/4! + 3637*x^5/5! + 55150*x^6/6! + 1033027*x^7/7! + 23260536*x^8/8! + 617066297*x^9/9! + 18968614874*x^10/10! + ...
such that
A(x) = 1 + (1 + exp(x))*x + (1 + exp(2*x))^2*x^2/2! + (1 + exp(3*x))^3*x^3/3! + (1 + exp(4*x))^4*x^4/4! + (1 + exp(5*x))^5*x^5/5! + (1 + exp(6*x))^6*x^6/6! + ...
also
A(x) = exp(x) + exp(x + exp(x)*x)*x + exp(4*x + exp(2*x)*x)*x^2/2! + exp(9*x + exp(3*x)*x)*x^3/3! + exp(16*x + exp(4*x)*x)*x^4/4! + exp(25*x + exp(5*x)*x)*x^5/5! + exp(36*x + exp(6*x)*x)*x^6/6! + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!  =  Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x ) * x^n/n!.
(1) At x = -1, the following sums are equal
S1 = Sum_{n>=0} (1 + exp(-n))^n * (-1)^n/n!,
S1 = Sum_{n>=0} exp(-n^2) * exp( -exp(-n) ) * (-1)^n/n!,
where S1 = 0.12121214669421724219987424741512642137552627624687959194...
(2) At x = -log(2), the following sums are equal
S2 = Sum_{n>=0} (1 + 1/2^n)^n * log(1/2)^n/n!,
S2 = Sum_{n>=0} 2^(-n^2) * 2^(-1/2^n) * log(1/2)^n/n!,
where S2 = 0.26746154600304489791062659014323146833150028333177021587...

Links

Crossrefs

Cf. A108459, A326091, A326261, A326009.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (1 + exp(n*x))^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (1 + 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 + exp(n*x)*x ) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m^2*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.: Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!.
E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x ) * x^n/n!.

A326550 E.g.f. A(x) satisfies: Sum_{n>=0} (exp(n*x) + A(x))^n * x^n / n! = Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!.

Original entry on oeis.org

1, 1, 3, 19, 183, 2451, 44523, 1003955, 27132591, 862477603, 31585560483, 1312509666051, 61199375982759, 3171627767105747, 181223609188848411, 11340823889035215187, 772846532507982245727, 57069311173494600701763, 4546598329397176113578067, 389300199395007408056468579, 35704214147724534934522349655, 3496767016630336049148287129971, 364696725110047554147731853993291
Offset: 0

Views

Author

Paul D. Hanna, Jul 13 2019

Keywords

Comments

RELATED IDENTITY: the following sums are equal:
(1) Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!.

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 183*x^4/4! + 2451*x^5/5! + 44523*x^6/6! + 1003955*x^7/7! + 27132591*x^8/8! + 862477603*x^9/9! + 31585560483*x^10/10! + 1312509666051*x^11/11! + ...
such that the following sums are equal
(1) B(x) = 1 + (exp(x) + A(x)) + (exp(2*x) + A(x))^2*x^2/2! + (exp(3*x) + A(x))^3*x^3/3! + (exp(4*x) + A(x))^4*x^4/4! + (exp(5*x) + A(x))^5*x^5/5! + ...
and
(2) B(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
(3) B(x) = exp(x*A(x)) + exp(x)*exp(exp(x)*x*A(x))*x + exp(4*x)*exp(exp(2*x)*x*A(x))*x^2/2! + exp(9*x)*exp(exp(3*x)*x*A(x))*x^3/3! + exp(16*x)*exp(exp(4*x)*x*A(x))*x^4/4! + ...
and
(4) B(x) = exp(x) + exp(x)*exp(exp(x)*x + x)*x + exp(4*x)*exp(exp(2*x)*x + 2*x)*x^2/2! + exp(9*x)*exp(exp(3*x)*x + 3*x)*x^3/3! + exp(16*x)*exp(exp(4*x)*x + 4*x)*x^4/4! + ...
where
B(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! + ... + A326009(n)*x^n/n! + ...

Links

Crossrefs

Cf. A326009.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=0,#A, (exp((m+1)*x +x*O(x^#A)) + 1)^m*x^m/m! - (exp(m*x +x*O(x^#A)) + Ser(A))^m*x^m/m! ),#A)); n!*A[n+1]}
for(n=0,25,print1(a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=0,#A, exp(m*(m+1)*x + exp(m*x +x*O(x^#A))*x )*x^m/m! - exp(m^2*x + exp(m*x +x*O(x^#A))*x * Ser(A))*x^m/m! ),#A)); n!*A[n+1]}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f. A(x) allows the following sums to be equal:
(1) B(x) = Sum_{n>=0} (exp(n*x) + A(x) )^n * x^n / n!,
(2) B(x) = Sum_{n>=0} (exp(n*x + x) + 1)^n * x^n / n!,
(3) B(x) = Sum_{n>=0} exp(n^2*x) * exp(exp(n*x)*x * A(x)) * x^n / n!,
(4) B(x) = Sum_{n>=0} exp(n^2*x) * exp(exp(n*x)*x + n*x ) * x^n / n!,
where B(x) is the e.g.f. of A326009.
a(n) = 1 (mod 2) for n >= 0.
a(6*n+1) = 2 (mod 3) for n >= 1;
a(6*n+3) = 1 (mod 3) for n >= 0;
a(6*n+k) = 0 (mod 3) when k = {0,2,4,5} for n >= 1.

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
Showing 1-4 of 4 results.

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