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.

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.

A326096 E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n!.

Original entry on oeis.org

1, 2, 6, 32, 256, 2712, 37744, 645752, 13371264, 327748832, 9332342944, 304875611328, 11298403070464, 470279355784448, 21809054992366464, 1118931830122060928, 63115145120561606656, 3892675200470654980608, 261242029823318546162176, 18994387868664467440590848, 1490356266852194536099393536, 125747158151444491631754033152
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 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 = 1.
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 = 1, r = x, m = 1.

Examples

			E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 32*x^3/3! + 256*x^4/4! + 2712*x^5/5! + 37744*x^6/6! + 645752*x^7/7! + 13371264*x^8/8! + 327748832*x^9/9! + 9332342944*x^10/10! + 304875611328*x^11/11! + 11298403070464*x^12/12! + ...
such that
A(x) = 1 + ((1+x) + 1)*x + ((1+x)^2 + 1)^2*x^2/2! + ((1+x)^3 + 1)^3*x^3/3! + ((1+x)^4 + 1)^4*x^4/4! + ((1+x)^5 + 1)^5*x^5/5! + ((1+x)^6 + 1)^6*x^6/6! + ((1+x)^7 + 1)^7*x^7/7! + ...
also
A(x) = 1 + (1+x)*exp(x*(1+x))*x + (1+x)^4*exp(x*(1+x)^2)*x^2/2! + (1+x)^9*exp(x*(1+x)^3)*x^3/3! + (1+x)^16*exp(x*(1+x)^4)*x^4/4! + (1+x)^25*exp(x*(1+x)^5)*x^5/5! + (1+x)^36*exp(x*(1+x)^6)*x^6/6! + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n!  =  Sum_{n>=0} (1+x)^(n^2) * exp(x*(1+x)^n) * x^n/n!.
(1) At x = -1/2, the following sums are equal
S1 = Sum_{n>=0} (-1)^n * 2^(-n*(n+1)) * (2^n + 1)^n / n!,
S1 = Sum_{n>=0} (-1)^n * 2^(-n*(n+1)) * exp( -1/2^(n+1) ) / n!,
where S1 = 0.41868678468707099609788224908427981408329845879700862624389...
(2) At x = -2/3, the following sums are equal
S2 = Sum_{n>=0} (-1)^n * 2^n * 3^(-n*(n+1)) * (3^n + 1)^n / n!,
S2 = Sum_{n>=0} (-1)^n * 2^n * 3^(-n*(n+1)) * exp( -2/3^(n+1) ) / n!,
where S2 = 0.33802063384093377391547056494398131361711992142768124149541...

Links

Crossrefs

Cf. A259208, A326092, A326093, A326094.

Programs

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

A326272 E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n * 2^n / n!.

Original entry on oeis.org

1, 2, 16, 264, 6736, 240160, 11214144, 657138944, 46862522368, 3973718103552, 393443889049600, 44826129808396288, 5806491899779117056, 846541984240702889984, 137723354275132587802624, 24818755539270666795663360, 4922319631768240931906584576, 1068365636390386171090826297344, 252495346180630403940163162472448, 64688594470052384103192832427687936, 17893635413553390198442202310639616000
Offset: 0

Views

Author

Paul D. Hanna, Jun 22 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 = -1, r = 2.
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 = -1, r = 2, m = 1.

Examples

			E.g.f: A(x) = 1 + 2*x + 16*x^2/2! + 264*x^3/3! + 6736*x^4/4! + 240160*x^5/5! + 11214144*x^6/6! + 657138944*x^7/7! + 46862522368*x^8/8! + 3973718103552*x^9/9! + 393443889049600*x^10/10! +...
such that
A(x) = 1 + 2*((1+x) - 1) + 2^2*((1+x)^2 - 1)^2/2! + 2^3*((1+x)^3 - 1)^3/3! + 2^4*((1+x)^4 - 1)^4/4! + 2^5*((1+x)^5 - 1)^5/5! + 2^6*((1+x)^6 - 1)^6/6! + 2^7*((1+x)^7 - 1)^7/7! + ...
also
A(x) = 1 + 2*(1+x)*exp(-2*(1+x)) + 2^2*(1+x)^4*exp(-2*(1+x)^2)/2! + 2^3*(1+x)^9*exp(-2*(1+x)^3)/3! + 2^4*(1+x)^16*exp(-2*(1+x)^4)/4! + 2^5*(1+x)^25*exp(-2*(1+x)^5)/5! + 2^6*(1+x)^36*exp(-2*(1+x)^6)/6! + 2^7*(1+x)^49*exp(-2*(1+x)^7)/7! + ...

Links

Crossrefs

Cf. A192935, A326273, A326274.
Cf. A326092.

Programs

  • PARI
    {a(n)=n!*polcoeff(sum(m=0, n, 2^m*((1+x+x*O(x^n))^m-1)^m/m!), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. may be expressed by the following sums.
(1) Sum_{n>=0} ((1+x)^n - 1)^n * 2^n / n!.
(2) Sum_{n>=0} (1+x)^(n^2) * exp(-2*(1+x)^n) * 2^n / n!.
Showing 1-4 of 4 results.

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