A181080 - OEIS

%I #5 Apr 05 2021 20:57:19

%S 1,1,2,4,14,83,774,10641,255918,14643874,1752083557,320079087261,

%T 79294841767020,27407454296637142,16895839815165609994,

%U 26064121763003372842186,82824096391548076720149081

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

%C Conjecture: this sequence consists entirely of integers.

%C Note that the following g.f. does NOT yield an integer series:

%C   exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k) * x^k] * x^n/n ).

%H G. C. Greubel, <a href="/A181080/b181080.txt">Table of n, a(n) for n = 0..90</a>

%e G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 +...

%e The logarithm of g.f. A(x) begins:

%e   log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + 4077*x^6/6 + ... + A181081(n)*x^n/n + ...

%e and equals the series:

%e   log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2

%e   + (1 + 3^3*x +  3^2*x^2 +      x^3)*x^3/3

%e   + (1 + 4^4*x +  6^3*x^2 +  4^2*x^3 +      x^4)*x^4/4

%e   + (1 + 5^5*x + 10^4*x^2 + 10^3*x^3 +  5^2*x^4 +     x^5)*x^5/5

%e   + (1 + 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...

%t 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 *)

%o (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)}

%o (Magma)

%o m:=20;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o 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

%o (Sage)

%o m=20;

%o def A181066_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     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()

%o A181066_list(m) # _G. C. Greubel_, Apr 05 2021

%Y Variants: A166894, A181070, A181082.

%Y Cf. A181081 (log).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 02 2010