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 1-3 of 3 results.

A240957 G.f.: Sum_{n>=0} n^n * x^n * (3 + 2*n*x)^n / ((1 + n*x)*(1 + 2*n*x))^(n+1).

Original entry on oeis.org

1, 3, 20, 234, 3944, 86400, 2324160, 74062800, 2726970624, 113893395840, 5319595814400, 274730601277440, 15544557784673280, 956232958853652480, 63540675378122342400, 4535620918350762240000, 346127227962539155292160, 28120835253815298895380480, 2423309442415144546546483200
Offset: 0

Views

Author

Paul D. Hanna, Aug 04 2014

Keywords

Examples

			O.g.f.: A(x) = 1 + 3*x + 20*x^2 + 234*x^3 + 3944*x^4 + 86400*x^5 +...
where
A(x) = 1 + x*(3+2*x)/((1+x)*(1+2*x))^2 + 2^2*x^2*(3+4*x)^2/((1+2*x)*(1+4*x))^3 + 3^3*x^3*(3+6*x)^3/((1+3*x)*(1+6*x))^4 + 4^4*x^4*(3+8*x)^4/((1+4*x)*(1+8*x))^5 + 5^5*x^5*(3+10*x)^5/((1+5*x)*(1+10*x))^6 +...

Links

Crossrefs

Cf. A240956, A240958, A240921.

Programs

  • Mathematica
    Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k, k] * 3^(n-2*k) * 2^k, {k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 05 2014 *)
  • PARI
    {a(n)=local(A=1); A=sum(m=0, n, m^m*x^m*(3+2*m*x)^m/((1 + m*x)*(1+2*m*x) +x*O(x^n))^(m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From formula for a(n): */
{Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!}
{a(n)=sum(k=0, n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k, k)*3^(n-2*k)*2^k)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * 3^(n-2*k) * 2^k.
a(n) ~ c * d^n * n! / sqrt(n), where d = 3*r^2/(2*r-1) + 2*(2*r-1)*r/(3*(1-r)) = 4.927267464017203368228591159442769988364645445182..., where r = 0.8093509687086163798199326301917112747442352555652682... is the root of the equation (r + 2*(1-2*r)^2/(9*(1-r))) * LambertW(-exp(-1/r)/r) = -1, and c = 0.546345652881951027770637598235474648132398514044679... . - Vaclav Kotesovec, Aug 05 2014

A240958 G.f.: Sum_{n>=0} n^n * x^n * (1 + 2*n*x)^n / (1 + n*x + 2*n^2*x^2)^(n+1).

Original entry on oeis.org

1, 1, 4, 30, 296, 3840, 60480, 1127280, 24240384, 590728320, 16090099200, 484387706880, 15971308784640, 572403619307520, 22155942961152000, 921115890645350400, 40935834850710159360, 1936630231160472207360, 97172886828612884889600, 5154401709528015200256000
Offset: 0

Views

Author

Paul D. Hanna, Aug 04 2014

Keywords

Comments

Generally, if Sum_{n>=0} n^n * x^n * (s + t*n*x)^n / (1 + s*n*x + t*n^2*x^2)^(n+1) = Sum_{n>=0} b(n)*x^n,
then b(n) = Sum_{k=0..n} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * s^(n-2*k) * t^k.
Limit n->infinity (b(n)/n!)^(1/n) = s*r^2/(2*r-1) + (2*r-1)*r*t/((1-r)*s), where r is the root of the equation (r + (1-2*r)^2 * t/((1-r)*s^2)) * LambertW(-exp(-1/r)/r) = -1. - Vaclav Kotesovec, Aug 05 2014

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 296*x^4 + 3840*x^5 + 60480*x^6 +...
where
A(x) = 1 + x*(1+2*x)/(1+x+2*x^2)^2 + 2^2*x^2*(1+4*x)^2/(1+2*x+8*x^2)^3 + 3^3*x^3*(1+6*x)^3/(1+3*x+18*x^2)^4 + 4^4*x^4*(1+8*x)^4/(1+4*x+32*x^2)^5 + 5^5*x^5*(1+10*x)^5/(1+5*x+50*x^2)^6 +...

Links

Crossrefs

Cf. A240956, A240957, A240921.

Programs

  • Mathematica
    Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k, k] * 2^k, {k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 05 2014 *)
  • PARI
    /* By a general formula for o.g.f.: */
{a(n,s,t)=local(A=1); A=sum(m=0, n, m^m*x^m*(s + t*m*x)^m/(1 + s*m*x + t*m^2*x^2 +x*O(x^n))^(m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n,1,2), ", "))
  • PARI
    /* By a general formula for a(n): */
{Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!}
{a(n,s,t)=sum(k=0, n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k, k)*s^(n-2*k)*t^k)}
for(n=0, 30, print1(a(n,1,2), ", "))

Formula

a(n) = Sum_{k=0..n} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * 2^k.
a(n) ~ c * d^n * n! / sqrt(n), where d = r^2/(2*r-1) + 2*(2*r-1)*r/(1-r) = 2.8672948250470036038473588196568091418984738141..., where r = 0.6842203847910787866923284795680321317882484098... is the root of the equation (r + 2*(1-2*r)^2/(1-r)) * LambertW(-exp(-1/r)/r) = -1, and c = 0.37767441309257908887250708986031213641309631613... . - Vaclav Kotesovec, Aug 05 2014

A240921 G.f.: Sum_{n>=0} n^n * x^n * (1 + n*x)^n / (1 + n*x + n^2*x^2)^(n+1).

Original entry on oeis.org

1, 1, 3, 18, 146, 1530, 19620, 297360, 5201784, 103146120, 2286181800, 56011087440, 1503057473280, 43844234353920, 1381310964633600, 46743301840435200, 1690919874777893760, 65116170597269151360, 2659604669692822051200, 114838104572526535200000, 5226654647185285702752000
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2014

Keywords

Comments

a(n) is divisible by [n/2]!.
Compare definition to the following identity, which holds for all fixed k:
Sum_{n>=0} n!*x^n = Sum_{n>=0} x^n * (n + k*x)^n / (1 + n*x + k*x^2)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 146*x^4 + 1530*x^5 + 19620*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2)^2 + 2^2*x^2*(1+2*x)^2/(1+2*x+4*x^2)^3 + 3^3*x^3*(1+3*x)^3/(1+3*x+9*x^2)^4 + 4^4*x^4*(1+4*x)^4/(1+4*x+16*x^2)^5 +...

Crossrefs

Cf. A240172, A240956, A240957, A240958, A240921, A008277 (Stirling2).

Programs

  • Mathematica
    Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Aug 03 2014 *)
  • PARI
    /* From definition: */
{a(n)=local(A=1); A=sum(m=0, n, m^m*x^m*(1+m*x)^m/(1 + m*x + m^2*x^2 +x*O(x^n))^(m+1)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From formula for a(n): */
{Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!}
{a(n)=sum(k=0,n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k,k))}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..[n/2]} (n-k)! * Stirling2(n, n-k) * binomial(n-k,k).
a(n) ~ c * d^n * n^n / exp(n), where d = r*(-1+3*r-3*r^2)/(1-3*r+2*r^2) = 2.334305682349197638435662..., where r = 0.722795640379451585372396... is the root of the equation (1-r) * (r + 1/LambertW(-exp(-1/r)/r)) + (2*r-1)^2 = 0, and c = 1.04764685950245700560418116727397... . - Vaclav Kotesovec, Aug 03 2014
Showing 1-3 of 3 results.

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