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A181084
Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} binomial(n,k)^(n+k+1) * x^k] * x^n/n ).
3
1, 1, 2, 10, 92, 1367, 87090, 20385333, 6633475836, 4096297538926, 14834973644512627, 119919823546238898903, 1273371038284317852447990, 41086272137585936052959008420, 6982122140549374036504235218052104
OFFSET
0,3
COMMENTS
Conjecture: this sequence consists entirely of integers.
LINKS
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...
The logarithm of g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + 513657*x^6/6 + ... + A181085(n)*x^n/n + ...
and equals the series:
log(A(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 + ...
MATHEMATICA
With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n+k+1)*x^(n+k)/n, {k, 0, n}], {n, m+1}]], {x, 0, m}], x]] (* G. C. Greubel, Apr 05 2021 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^(m+k+1)*x^k)*x^m/m) + x*O(x^n)), n)}
(Magma)
m:=20;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n, k)^(n+k+1)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
(Sage)
m=20;
def A181084_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( exp( sum( sum( binomial(n, k)^(n+k+1)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
A181084_list(m) # G. C. Greubel, Apr 05 2021
CROSSREFS
Cf. A181085 (log), variants: A181080, A181082.
Sequence in context: A289020 A195415 A336271 * A063385 A293709 A063393
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 28 2010
STATUS
approved