A181083 -id:A181083 - cp's OEIS frontend

A181082 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n+k) * x^k] * x^n/n ).

1, 1, 2, 6, 34, 375, 15200, 2066401, 450054919, 199271253643, 431399012916702, 2151987403947457136, 15451465958263071713102, 331187643758039140349444047, 33475597220485400781283541412048

Offset: 0

Paul D. Hanna, Oct 02 2010

nonn

Conjecture: this sequence consists entirely of integers.

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 34*x^4 + 375*x^5 + 15200*x^6 +...
The logarithm of g.f. A(x) begins:
  log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 111*x^4/4 + 1686*x^5/5 + 88737*x^6/6 + ... + A181083(n)*x^n/n + ...
and equals the series:
  log(A(x)) = (1 + x)*x + (1 + 2^3*x + x^2)*x^2/2
  + (1 + 3^4*x +  3^5*x^2 +      x^3)*x^3/3
  + (1 + 4^5*x +  6^6*x^2 +  4^7*x^3 +       x^4)*x^4/4
  + (1 + 5^6*x + 10^7*x^2 + 10^8*x^3 +   5^9*x^4 +      x^5)*x^5/5
  + (1 + 6^7*x + 15^8*x^2 + 20^9*x^3 + 15^10*x^4 + 6^11*x^5 + x^6)*x^6/6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..70

Cf. A181083 (log), variants: A181080, A181084.

m:=20;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(n+k)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021

With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n+k)*x^(n+k)/n, {k, 0,n}], {n,m+1}]], {x,0,m}], x]] (* G. C. Greubel, Apr 05 2021 *)

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k)*x^k)*x^m/m)+x*O(x^n)), n)}

m=20;
def A181082_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n,k)^(n+k)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
A181082_list(m) # G. C. Greubel, Apr 05 2021

A181085 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n+1) * n/(n-k).

Original entry on oeis.org

1, 3, 25, 327, 6336, 513657, 142074241, 52903930911, 36806786795365, 148308705637730728, 1318954828711012426638, 15279013243159345043036553, 534104982404807772659968455891, 97749134742042348389685885848315523

Offset: 1

Paul D. Hanna, Oct 28 2010

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + ...
which equals the series:
L(x) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2
+ (1+ 3^5*x + 3^6*x^2 + x^3)*x^3/3
+ (1+ 4^6*x + 6^7*x^2 + 4^8*x^3 + x^4)*x^4/4
+ (1+ 5^7*x + 10^8*x^2 + 10^9*x^3 + 5^10*x^4 + x^5)*x^5/5
+ (1+ 6^8*x + 15^9*x^2 + 20^10*x^3 + 15^11*x^4 + 6^12*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181084:
exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 1..60

Variants: A166895, A181071, A181081, A181083.

Cf. A181084 (exp).

[(&+[Binomial(n-j,j)^(n+1)*(n/(n-j)): j in [0..Floor(n/2)]]): j in [1..20]]; // G. C. Greubel, Apr 04 2021

Table[Sum[Binomial[n-k, k]^(n+1)*(n/(n-k)), {k, 0, Floor[n/2]}], {n, 20}] (* G. C. Greubel, Apr 04 2021 *)

a(n)=sum(k=0, n\2, binomial(n-k, k)^(n+1)*n/(n-k))

{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k+1)*x^k)*x^m/m)+x*O(x^n), n)}

[sum( binomial(n-k, k)^(n+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021

cp's OEIS Frontend

A181082 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n+k) * x^k] * x^n/n ).

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A181085 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n+1) * n/(n-k).

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Magma

Mathematica

PARI

PARI

Sage