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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 1-2 of 2 results.

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.

A276747 G.f.: Sum_{n>=0} (1-x)^(n*(n+1)) * [ Sum_{k>=1} k^n * x^k ]^n.

Original entry on oeis.org

1, 1, 1, 3, 14, 96, 989, 14264, 293081, 8291372, 326486284, 17606371379, 1311003529532, 133789640100606, 18842361596022104, 3651812223033372781, 979595054829206809506, 363619011980801177687068, 187594865162514096249150130, 134684579087971548803896902904, 134956937109764143572996094860839, 189135846049140695927044178145555683, 371258683769470709816610430835777163052
Offset: 0

Views

Author

Paul D. Hanna, Sep 30 2016

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 96*x^5 + 989*x^6 + 14264*x^7 + 293081*x^8 + 8291372*x^9 + 326486284*x^10 + 17606371379*x^11 +...
such that
A(x) = Sum_{n>=0} (1-x)^(n*(n+1)) * (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n.
Explicitly,
A(x) = 1 + (1-x)^2 * (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 +...) +
(1-x)^6 * (x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 +...)^2 +
(1-x)^12 * (x + 8*x^2 + 27*x^3 + 64*x^4 + 125*x^5 + 216*x^6 +...)^3 +
(1-x)^20 * (x + 16*x^2 + 81*x^3 + 256*x^4 + 625*x^5 + 1296*x^6 +...)^4 +
(1-x)^30 * (x + 32*x^2 + 243*x^3 + 1024*x^4 + 3125*x^5 + 7776*x^6 +...)^5 +
...
The g.f. can be written using the Eulerian numbers like so:
A(x) = 1 + x + (x + x^2)^2 + (x + 4*x^2 + x^3)^3 + (x + 11*x^2 + 11*x^3 + x^4)^4 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6 + (x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7)^7 + (x + 247*x^2 + 4293*x^3 + 15619*x^4 + 15619*x^5 + 4293*x^6 + 247*x^7 + x^8)^8 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n +...

Links

Crossrefs

Cf. A276743, A276746, A008292.

Programs

  • PARI
    {a(n) = my(A=1); A = sum(m=0,n+1, (1-x +x*O(x^n))^(m*(m+1)) * sum(k=1,n+1,k^m*x^k +x*O(x^n))^m );polcoeff(A,n)}
for(n=0, 30, 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, sum(k=1, m, A008292(m, k)*x^k +Oxn )^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

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

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