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.

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!.

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!.

A326274 E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n * 4^n / n!.

Original entry on oeis.org

1, 4, 64, 1920, 86464, 5304320, 418131456, 40727959552, 4765747597312, 655794545577984, 104360850604687360, 18948720298674028544, 3882059495694122090496, 889053986706845142876160, 225799026538694916941283328, 63163063632830911303738982400, 19344290761718462120859544846336, 6452149866509553556278434299117568, 2332867461867950308492384248149311488, 910538103145382496893587688740637114368, 382208425560563535419125500691963382333440
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 = 4.
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 = 4, m = 1.

Examples

			E.g.f: A(x) = 1 + 4*x + 64*x^2/2! + 1920*x^3/3! + 86464*x^4/4! + 5304320*x^5/5! + 418131456*x^6/6! + 40727959552*x^7/7! + 4765747597312*x^8/8! + 655794545577984*x^9/9! + 104360850604687360*x^10/10! +...
such that
A(x) = 1 + 4*((1+x) - 1) + 4^2*((1+x)^2 - 1)^2/2! + 4^3*((1+x)^3 - 1)^3/3! + 4^4*((1+x)^4 - 1)^4/4! + 4^5*((1+x)^5 - 1)^5/5! + 4^6*((1+x)^6 - 1)^6/6! + 4^7*((1+x)^7 - 1)^7/7! + ...
also
A(x) = 1 + 4*(1+x)*exp(-4*(1+x)) + 4^2*(1+x)^4*exp(-4*(1+x)^2)/2! + 4^3*(1+x)^9*exp(-4*(1+x)^3)/3! + 4^4*(1+x)^16*exp(-4*(1+x)^4)/4! + 4^5*(1+x)^25*exp(-4*(1+x)^5)/5! + 4^6*(1+x)^36*exp(-4*(1+x)^6)/6! + 4^7*(1+x)^49*exp(-4*(1+x)^7)/7! + ...

Links

Crossrefs

Cf. A192935, A326272, A326273.
Cf. A326094.

Programs

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

A192985 E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n/n!.

Original entry on oeis.org

1, 1, 2, 18, 324, 8600, 304110, 13494012, 723167816, 45583507944, 3314590951050, 273983096442440, 25451868722986332, 2633115724586963772, 301154828427859401086, 37850982841326873432060, 5202124730575982650388880
Offset: 0

Views

Author

Paul D. Hanna, Jul 13 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 18*x^3/3! + 324*x^4/4! + 8600*x^5/5! +...
where A(x/G(x)) = 1 + x and G(x) is the e.g.f. of A192935:
G(x) = 1 + x + 4*x^2/2! + 39*x^3/3! + 592*x^4/4! + 12965*x^5/5! +...

Crossrefs

Cf. A192935.

Formula

E.g.f. satisfies: A(x/G(x)) = 1 + x where G(x) = Sum_{n>=0} ((1+x)^n-1)^n/n! and G(x) = x/Series_Reversion(A(x)-1) = e.g.f. of A192935.
Showing 1-4 of 4 results.

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