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 1-2 of 2 results.

A337852 a(n) = (2^(n+1) + 1)^n.

Original entry on oeis.org

1, 5, 81, 4913, 1185921, 1160290625, 4608273662721, 74051159531521793, 4796659837465472798721, 1248862969947666168212890625, 1304426412609681656861792686592001, 5459157240288132828933147334116110282753, 91477746675481294892349178081259839233191936001
Offset: 0

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Author

Paul D. Hanna, Sep 26 2020

Keywords

Comments

In general, we have the o.g.f. identity:
Sum_{n>=0} m^n * q^(n^2) * x^n/(1 - b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n ; here, q=2, m=2, b=1.
In general, we have the e.g.f. identity:
Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! = Sum_{n>=0} (m*q^n + b)^n * x^n / n! ; here, q=2, m=2, b=1.

Examples

			O.g.f.: A(x) = 1 + 5*x + 81*x^2 + 4913*x^3 + 1185921*x^4 + 1160290625*x^5 + 4608273662721*x^6 + 74051159531521793*x^7 + 4796659837465472798721*x^8 + ...
where
A(x) = 1/(1 - x) + 2^2*x/(1 - 2*x)^2 + 2^6*x^2/(1 - 2^2*x)^3 + 2^12*x^3/(1 - 2^3*x)^4 + 2^20*x^4/(1 - 2^4*x)^5 + 2^30*x^5/(1 - 2^5*x)^6 + ...

Crossrefs

Cf. A055601, A251657, A337851, A136516.

Programs

  • PARI
    {a(n,q,m,b) = (m*q^n + b)^n}
for(n=0,15, print1(a(n,q=2,m=2,b=1),", "))
  • PARI
    /* E.g.f. formula: */
{a(n,q,m,b) = polcoeff( sum(k=0,n, m^k * q^(k^2) * x^k / (1 - b*q^k*x +x*O(x^n))^(k+1)), n)}
for(n=0,15, print1(a(n,q=2,m=2,b=1),", "))
  • PARI
    /* E.g.f. formula: */
{a(n,q,m,b) = n! * polcoeff( sum(k=0,n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)}
for(n=0,15, print1(a(n,q=2,m=2,b=1),", "))

Formula

O.g.f.: Sum_{n>=0} 2^(n*(n+1)) * x^n/(1 - 2^n*x)^(n+1) = Sum_{n>=0} (2^(n+1) + 1)^n * x^n.
E.g.f.: Sum_{n>=0} 2^(n*(n+1)) * exp(2^n*x) * x^n / n! = Sum_{n>=0} (2^(n+1) + 1)^n * x^n / n!.

A341378 G.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n*(1 + x*A(x)^n)^n and Q(x) = Sum_{n>=0} x^n*(1 + x*A(x)^n)^n.

Original entry on oeis.org

1, 1, 2, 4, 13, 49, 198, 847, 3796, 17661, 84754, 417556, 2104652, 10825545, 56714218, 302185765, 1635727560, 8987357912, 50090647729, 283058794945, 1621221821820, 9409181197452, 55327487031279, 329591827614844, 1989063796779401
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2021

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 13*x^4 + 49*x^5 + 198*x^6 + 847*x^7 + 3796*x^8 + 17661*x^9 + 84754*x^10 + 417556*x^11 + 2104652*x^12 + ...
such that A(x) = P(x)/Q(x) where
P(x) = 1 + 2*x*(1 + x*A(x)) + 3*x^2*(1 + x*A(x)^2)^2 + 4*x^3*(1 + x*A(x)^3)^3 + 5*x^4*(1 + x*A(x)^4)^4 + 6*x^5*(1 + x*A(x)^5)^5 + ...
Q(x) = 1 + x*(1 + x*A(x)) + x^2*(1 + x*A(x)^2)^2 + x^3*(1 + x*A(x)^3)^3 + x^4*(1 + x*A(x)^4)^4 + x^5*(1 + x*A(x)^5)^5 + ...
also
Q(x) = 1/(1 - x) + x^2*A(x)/(1 - x*A(x))^2 + x^4*A(x)^4/(1 - x*A(x)^2)^3 + x^6*A(x)^9/(1 - x*A(x)^3)^4 + x^8*A(x)^16/(1 - x*A(x)^4)^5 + ...
explicitly,
P(x) = 1 + 2*x + 5*x^2 + 12*x^3 + 36*x^4 + 124*x^5 + 471*x^6 + 1918*x^7 + 8246*x^8 + 37024*x^9 + 172266*x^10 + 825870*x^11 + 4062450*x^12 + ...
Q(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 35*x^5 + 125*x^6 + 484*x^7 + 1994*x^8 + 8625*x^9 + 38804*x^10 + 180360*x^11 + 861833*x^12 + ...

Crossrefs

Cf. A337851.

Programs

  • PARI
    {a(n) = my(A=1 +x*O(x^n)); for(i=1,n,
A = sum(m=0,n, (m+1) * x^m *(1 + x*A^m +x*O(x^n))^m ) / sum(m=0,n, x^m *(1 + x*A^m +x*O(x^n))^m )   );
polcoeff(H=A,n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = P(x)/Q(x) where
P(x) = Sum_{n>=0} (n+1) * x^n * (1 + x*A(x)^n)^n and
Q(x) = Sum_{n>=0} x^n * (1 + x*A(x)^n)^n.
(2) A(x) = P(x)/Q(x) where
P(x) = Sum_{n>=0} (n+1) * x^n * (1 + x*A(x)^n)^n and
Q(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2) / (1 - x*A^n)^(n+1).
Showing 1-2 of 2 results.

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