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.

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!.

A326273 E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n * 3^n / n!.

Original entry on oeis.org

1, 3, 36, 837, 29592, 1439775, 90723564, 7109399241, 672900166584, 75245901590187, 9770338275393240, 1452674820992915817, 244491148094925021156, 46131995287645828742727, 9678693008639052537757380, 2241968557540165237891804185, 569848346606872473737714179056, 158069419606634839915503628956051, 47621655849844748263169576451111984, 15515379326590122849811694557147948473, 5445580659887211921286711773580373201820
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 = 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 = -1, r = 3, m = 1.

Examples

			E.g.f: A(x) = 1 + 3*x + 36*x^2/2! + 837*x^3/3! + 29592*x^4/4! + 1439775*x^5/5! + 90723564*x^6/6! + 7109399241*x^7/7! + 672900166584*x^8/8! + 75245901590187*x^9/9! + 9770338275393240*x^10/10! +...
such that
A(x) = 1 + 3*((1+x) - 1) + 3^2*((1+x)^2 - 1)^2/2! + 3^3*((1+x)^3 - 1)^3/3! + 3^4*((1+x)^4 - 1)^4/4! + 3^5*((1+x)^5 - 1)^5/5! + 3^6*((1+x)^6 - 1)^6/6! + 3^7*((1+x)^7 - 1)^7/7! + ...
also
A(x) = 1 + 3*(1+x)*exp(-3*(1+x)) + 3^2*(1+x)^4*exp(-3*(1+x)^2)/2! + 3^3*(1+x)^9*exp(-3*(1+x)^3)/3! + 3^4*(1+x)^16*exp(-3*(1+x)^4)/4! + 3^5*(1+x)^25*exp(-3*(1+x)^5)/5! + 3^6*(1+x)^36*exp(-3*(1+x)^6)/6! + 3^7*(1+x)^49*exp(-3*(1+x)^7)/7! + ...

Links

Crossrefs

Cf. A192935, A326272, A326274.
Cf. A326093.

Programs

  • PARI
    {a(n)=n!*polcoeff(sum(m=0, n, 3^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 * 3^n / n!.
(2) Sum_{n>=0} (1+x)^(n^2) * exp(-3*(1+x)^n) * 3^n / n!.
Showing 1-4 of 4 results.

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