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.

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A277036 G.f.: Sum_{n>=0} exp(-n * 2^n * x) * [ Sum_{k>=1} k^n * 2^(n*k) * x^k / k! ]^n.

Original entry on oeis.org

1, 2, 16, 640, 102656, 63897600, 154597064704, 1463095704682496, 54479037904873062400, 8016231806154061580861440, 4675328432258454936484990418944, 10830326782491721013522399339743281152, 99782643106894570834269165391541758337220608, 3659836060539105945122413831815090863199825623515136, 534751190090057629985959636400471838795213939324687126364160
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2016

Keywords

Comments

More generally, for fixed integer q, G(x,q) = Sum_{n>=0} exp(-n * q^n * x) * [ Sum_{k>=1} k^n * q^(n*k) * x^k / k! ]^n is an integer series such that G(x,q) = Sum_{n>=0} q^(n^2) * [ Sum_{k=1..n} S2(n,k) * q^(n*k-n) * x^k ]^n.

Examples

			G.f.: A(x) = 1 + 2*x + 16*x^2 + 640*x^3 + 102656*x^4 + 63897600*x^5 + 154597064704*x^6 + 1463095704682496*x^7 +...
such that
A(x) = Sum_{n>=0} exp(-n*2^n*x) * (2^n*x + 2^n*2^(2*n)*x^2/2! + 3^n*2^(3*n)*x^3/3! +...+ k^n*2^(k*n)*x^k/k! +...)^n.
Explicitly,
A(x) = 1 + exp(-2*x) * (2*x + 2*2^2*x^2/2! + 3*2^3*x^3/3! + 4*2^4*x^4/4! +...) +
exp(-2*2^2*x) * (2^2*x + 4*2^4*x^2/2! + 9*2^6*x^3/3! + 16*2^8*x^4/4! +...)^2 +
exp(-3*2^3*x) * (2^3*x + 8*2^6*x^2/2! + 27*2^9*x^3/3! + 64*2^12*x^4/4! +...)^3 +
exp(-4*2^4*x) * (2^4*x + 16*2^8*x^2/2! + 81*2^12*x^3/3! + 256*2^16*x^4/4! +...)^4 +
exp(-5*2^5*x) * (2^5*x + 32*2^10*x^2/2! + 243*2^15*x^3/3! + 1024*2^20*x^4/4! +...)^5 +...
The g.f. can be written using the Stirling2 numbers like so:
A(x) = 1 + 2*x + (2^2*x + 2^4*x^2)^2 + (2^3*x + 3*2^6*x^2 + 2^9*x^3)^3 + (2^4*x + 7*2^8*x^2 + 6*2^12*x^3 + 2^16*x^4)^4 + (2^5*x + 15*2^10*x^2 + 25*2^15*x^3 + 10*2^20*x^4 + 2^25*x^5)^5 + (2^6*x + 31*2^12*x^2 + 90*2^18*x^3 + 65*2^24*x^4 + 15*2^30*x^5 + 2^36*x^6)^6 + (2^7*x + 63*2^14*x^2 + 301*2^21*x^3 + 350*2^28*x^4 + 140*2^35*x^5 + 21*2^42*x^6 + 2^49*x^7)^7 +...+ [ Sum_{k=1..n} S2(n,k) * 2^(n*k) * x^k ]^n +...

Crossrefs

Cf. A276746, A277037, A168407.

Programs

  • PARI
    {a(n) = my(A=1); A = sum(m=0, n+1, exp(-m*2^m*x +x*O(x^n)) * sum(k=1, n+1, 2^(m*k)*k^m*x^k/k! +x*O(x^n))^m ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
  • PARI
    {a(n) = my(A = sum(m=0, n, sum(k=1, m, stirling(m, k, 2)*2^(m*k)*x^k +x*O(x^n) )^m )); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} [ Sum_{k=1..n} S2(n,k) * 2^(n*k) * x^k ]^n, where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.

A277035 G.f.: Sum_{n>=0} log(1 + Sum_{k>=1} k^n * 2^(n*k) * x^k )^n / n!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 14, 320, 21036, 4248736, 2753284608, 5889659239296, 42571528094271584, 1060699597956427433984, 92622122614950875482410496, 28732153964162783015337150191616, 32013867511269166370870196132112760832, 129287051721999031624124705228031781712207872, 1906782843976072893849368486957954962408685271556096
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2016

Keywords

Comments

More generally, for fixed integer q, G(x,q) = Sum_{n>=0} log(1 + Sum_{k>=1} k^n * q^(n*k) * x^k )^n / n! is an integer series such that G(x,q) = Sum_{n>=0} log(1 + [Sum_{k=1..n} A008292(n,k) * q^(n*k) * x^k]/(1 - q^n*x)^(n+1) )^n / n!, where A008292 are the Eulerian numbers.

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 320*x^3 + 21036*x^4 + 4248736*x^5 + 2753284608*x^6 + 5889659239296*x^7 + 42571528094271584*x^8 +...
such that
A(x) = Sum_{n>=0} log(1 + 1^n*2^n*x + 2^n*2^(2*n)*x^2 + 3^n*2^(3*n)*x^3 +...+ k^n*2^(k*n)*x^k +...)^n/n!.
Equivalently,
A(x) = 1 + log(1 + 1*2*x + 2*2^2*x^2 + 3*2^3*x^3 + 4*2^4*x^4 +...) +
log(1 + 1^2*2^2*x + 2^2*2^4*x^2 + 3^2*2^6*x^3 + 4^2*2^8*x^4 +...)^2/2! +
log(1 + 1^3*2^3*x + 2^3*2^6*x^2 + 3^3*2^9*x^3 + 4^3*2^12*x^4 +...)^3/3! +
log(1 + 1^4*2^4*x + 2^4*2^8*x^2 + 3^4*2^12*x^3 + 4^4*2^16*x^4 +...)^4/4! +
...
The g.f. can be written using the Eulerian numbers like so:
A(x) = 1 + log(1 + 2*x/(1-2*x)^2) + log(1 + 2^2*(x + 2^2*x^2)/(1-2^2*x)^3)^2/2! + log(1 + 2^3*(x + 4*2^3*x^2 + 2^6*x^3)/(1-2^3*x)^4)^3/3! + log(1 + 2^4*(x + 11*2^4*x^2 + 11*2^8*x^3 + 2^24*x^4)/(1-2^4*x)^5)^4/4! + log(1 + 2^5*(x + 26*2^5*x^2 + 66*2^10*x^3 + 26*2^15*x^4 + 2^20*x^5)/(1-2^5*x)^6)^5/5! + log(1 + 2^6*(x + 57*2^6*x^2 + 302*2^12*x^3 + 302*2^18*x^4 + 57*2^24*x^5 + 2^30*x^6)/(1-2^6*x)^7)^6/6! +...+ log(1 + Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k / (1 - 2^n*x)^(n+1) )^n/n! +...

Crossrefs

f. A277037, A277036, A008292.

Programs

  • PARI
    {a(n) = my(A=1, Oxn=x*O(x^n));
A = sum(m=0, n+1, log(1 + sum(k=1, n+1, k^m * 2^(m*k) * x^k +x*O(x^n)) )^m / m! );
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n));
A = sum(m=0, n+1, log(1 + sum(k=1, m+1, A008292(m,k) * 2^(m*k) * x^k) / (1 - 2^m*x +Oxn)^(m+1) )^m / m! );
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} log(1 + [Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k]/(1 - 2^n*x)^(n+1) )^n / n!, where A008292 are the Eulerian numbers.
Showing 1-2 of 2 results.

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