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.

A326092 E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!.

Original entry on oeis.org

1, 3, 11, 63, 525, 5883, 84519, 1494783, 31854489, 800205075, 23315862339, 777867156927, 29384670476709, 1245177345486987, 58718905551858015, 3060140159517853887, 175176443950054714161, 10955959246057628397987, 745058168844977314910331, 54857350105041217492956735, 4356213264604432880789346621
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2019

Keywords

Comments

More generally, 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!;
here, q = (1+x) and p = 2, r = x.
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = 1+x, p = 2, r = x, m = 1.

Examples

			E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 63*x^3/3! + 525*x^4/4! + 5883*x^5/5! + 84519*x^6/6! + 1494783*x^7/7! + 31854489*x^8/8! + 800205075*x^9/9! + 23315862339*x^10/10! + ...
such that
A(x) = 1 + ((1+x) + 2)*x + ((1+x)^2 + 2)^2*x^2/2! + ((1+x)^3 + 2)^3*x^3/3! + ((1+x)^4 + 2)^4*x^4/4! + ((1+x)^5 + 2)^5*x^5/5! + ((1+x)^6 + 2)^6*x^6/6! + ((1+x)^7 + 2)^7*x^7/7! + ...
also
A(x) = 1 + (1+x)*exp(2*x*(1+x))*x + (1+x)^4*exp(2*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(2*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(2*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(2*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(2*x*(1+x)^6)*x^6/6! + ...

Links

Crossrefs

Cf. A326096, A326093, A326094.
Cf. A326272.

Programs

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

Formula

E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!,
E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(2*x*(1+x)^n) * x^n/n!.
a(n) = 0 (mod 3) for n > 2.

A326093 E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n!.

Original entry on oeis.org

1, 4, 18, 112, 976, 11424, 169936, 3101032, 67876608, 1746757504, 52034505376, 1771434644544, 68180144988928, 2939951026982272, 140920461751138176, 7457658363325181824, 433145750643704774656, 27464893679743640343552, 1892311278990953945563648, 141074242336048184406390784, 11336870115013701213795557376
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} (q^n + p)^n * x^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*x) * x^n/n!;
here, q = (1+x) and p = 3.
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = 1+x, p = 3, r = x, m = 1.

Examples

			E.g.f.: A(x) = 1 + 4*x + 18*x^2/2! + 112*x^3/3! + 976*x^4/4! + 11424*x^5/5! + 169936*x^6/6! + 3101032*x^7/7! + 67876608*x^8/8! + 1746757504*x^9/9! + 52034505376*x^10/10! + ...
such that
A(x) = 1 + ((1+x) + 3)*x + ((1+x)^2 + 3)^2*x^2/2! + ((1+x)^3 + 3)^3*x^3/3! + ((1+x)^4 + 3)^4*x^4/4! + ((1+x)^5 + 3)^5*x^5/5! + ((1+x)^6 + 3)^6*x^6/6! + ((1+x)^7 + 3)^7*x^7/7! + ...
also
A(x) = 1 + (1+x)*exp(3*x*(1+x))*x + (1+x)^4*exp(3*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(3*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(3*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(3*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(3*x*(1+x)^6)*x^6/6! + ...

Links

Crossrefs

Cf. A326096, A326092, A326094.
Cf. A326273.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n! */
{a(n) = my(A = sum(m=0,n, ((1+x)^m + 3 +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} (1+x)^(n^2) * exp(3*x*(1+x)^n) * x^n/n! */
{a(n) = my(A = sum(m=0,n, (1+x +x*O(x^n))^(m^2) * exp(3*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n!,
E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(3*x*(1+x)^n) * x^n/n!.
a(n) = 0 (mod 4) for n > 2.

A326094 E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n!.

Original entry on oeis.org

1, 5, 27, 185, 1693, 20565, 316375, 5948465, 133579065, 3517749125, 107024710675, 3714813650025, 145570443534805, 6383184292589525, 310815510350462415, 16694390352153656225, 983323269272332915825, 63186890982241624232325, 4409134435821084657726475, 332714992062735780407411225
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2019

Keywords

Comments

More generally, 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*x) * r^n/n!;
here, q = (1+x) and p = 4, r = x.
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = 1+x, p = 4, r = x, m = 1.

Examples

			E.g.f.: A(x) = 1 + 5*x + 27*x^2/2! + 185*x^3/3! + 1693*x^4/4! + 20565*x^5/5! + 316375*x^6/6! + 5948465*x^7/7! + 133579065*x^8/8! + 3517749125*x^9/9! + 107024710675*x^10/10! + ...
such that
A(x) = 1 + ((1+x) + 4)*x + ((1+x)^2 + 4)^2*x^2/2! + ((1+x)^3 + 4)^3*x^3/3! + ((1+x)^4 + 4)^4*x^4/4! + ((1+x)^5 + 4)^5*x^5/5! + ((1+x)^6 + 4)^6*x^6/6! + ((1+x)^7 + 4)^7*x^7/7! + ...
also
A(x) = 1 + (1+x)*exp(4*x*(1+x))*x + (1+x)^4*exp(4*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(4*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(4*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(4*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(4*x*(1+x)^6)*x^6/6! + ...

Links

Crossrefs

Cf. A326096, A326092, A326093.
Cf. A326274.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n! */
{a(n) = my(A = sum(m=0,n, ((1+x)^m + 4 +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} (1+x)^(n^2) * exp(4*x*(1+x)^n) * x^n/n! */
{a(n) = my(A = sum(m=0,n, (1+x +x*O(x^n))^(m^2) * exp(4*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n!,
E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(4*x*(1+x)^n) * x^n/n!.
a(n) = 0 (mod 5) for n > 4.

A259208 E.g.f.: Sum_{n>=0} x^n * (1 + x^n)^n / n!.

Original entry on oeis.org

1, 1, 3, 1, 25, 1, 721, 1, 6721, 181441, 151201, 1, 203575681, 1, 121080961, 108972864001, 3491282995201, 1, 133541574566401, 1, 304119455447808001, 212878925715456001, 309744468633601, 1, 17665560315112008499201, 646300418472124416000001, 841941782922240001
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2015

Keywords

Comments

More generally, 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!;
here, q = x, p = 1, r = x. - Paul D. Hanna, Jul 03 2019
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = , p = 1, r = x, m = 1. - Paul D. Hanna, Jul 06 2019

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + x^3/3! + 25*x^4/4! + x^5/5! + 721*x^6/6! + x^7/7! + 6721*x^8/8! + 181441*x^9/9! +...
where
A(x) = 1 + x*(1+x) + x^2*(1+x^2)^2/2! + x^3*(1+x^3)^3/3! + x^4*(1+x^4)^4/4! + x^5*(1+x^5)^5/5! + x^6*(1+x^6)^6/6! +...
also
A(x) = exp(x) + x^2*exp(x^2) + x^6*exp(x^3)/2! + x^12*exp(x^4)/3! + x^20*exp(x^5)/4! + x^30*exp(x^6)/5! + x^42*exp(x^7)/6! +...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} x^n * (1 + x^n)^n / n!  =  Sum_{n>=0} x^(n*(n+1)) * exp(x^(n+1)) / n!.
(1) At x = 1/2, the following sums are equal
S1 = Sum_{n>=0} 2^(-n*(n+1)) * (2^n + 1)^n / n!,
S1 = Sum_{n>=0} 2^(-n*(n+1)) * exp( 1/2^(n+1) ) / n!,
where S1 = 1.97862370255774939923047215233920359461142155926360261512472...
(2) At x = 1/3, the following sums are equal
S2 = Sum_{n>=0} 3^(-n*(n+1)) * (3^n + 1)^n / n!,
S2 = Sum_{n>=0} 3^(-n*(n+1)) * exp( 1/3^(n+1) ) / n!,
where S2 = 1.52049327799122758174016893855657751957768595647287750170026...

Links

Crossrefs

Cf. A326096.

Programs

  • PARI
    {a(n) = local(A=1); A = sum(m=0, n, x^m/m!*(1 + x^m +x*O(x^n))^m); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=1); A = sum(m=0, n, x^(m*(m+1)) / m! * exp(x^(m+1) +x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = if(n==0,1, sumdiv(n,d, binomial(d, n/d-1) * n!/d! ) )}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: Sum_{n>=0} x^(n*(n+1)) * exp(x^(n+1)) / n!.
a(n) = Sum_{d|n} binomial(d, n/d - 1) * n!/d!.
Showing 1-4 of 4 results.

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