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A181080 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k+1) * x^k] * x^n/n ). 6
1, 1, 2, 4, 14, 83, 774, 10641, 255918, 14643874, 1752083557, 320079087261, 79294841767020, 27407454296637142, 16895839815165609994, 26064121763003372842186, 82824096391548076720149081 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Conjecture: this sequence consists entirely of integers.
Note that the following g.f. does NOT yield an integer series:
exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k) * x^k] * x^n/n ).
LINKS
EXAMPLE
G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 +...
The logarithm of g.f. A(x) begins:
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 + ...
and equals the series:
log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
+ (1 + 3^3*x + 3^2*x^2 + x^3)*x^3/3
+ (1 + 4^4*x + 6^3*x^2 + 4^2*x^3 + x^4)*x^4/4
+ (1 + 5^5*x + 10^4*x^2 + 10^3*x^3 + 5^2*x^4 + x^5)*x^5/5
+ (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 + ...
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 A181066_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()
A181066_list(m) # G. C. Greubel, Apr 05 2021
CROSSREFS
Variants: A166894, A181070, A181082.
Cf. A181081 (log).
Sequence in context: A093462 A302136 A371674 * A166894 A232108 A339651
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 02 2010
STATUS
approved

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Last modified June 19 13:40 EDT 2024. Contains 373503 sequences. (Running on oeis4.)