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A181074 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(k+1) *x^k ] *x^n/n ). 6
1, 1, 2, 5, 23, 231, 5405, 322799, 42761356, 12597156231, 9136063939651, 14655841196011960, 51639276405198967750, 449212631407010945983244, 8871353886432410987179493370, 378793180251425841753491012596531 (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} C(n+k-1,k)^k * x^k] * x^n/n ).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..75

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 +...

The logarithm begins:

log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*x^5/5 + 30912*x^6/6 +...+ A181075(n)*x^n/n +...

which equals the series:

  log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x

  + (1 + 2^2*x +  3^3*x^2 +  4^4*x^3 +   5^5*x^4 +   6^6*x^5 + ...)*x^2/2

  + (1 + 3^2*x +  6^3*x^2 + 10^4*x^3 +  15^5*x^4 +  21^6*x^5 + ...)*x^3/3

  + (1 + 4^2*x + 10^3*x^2 + 20^4*x^3 +  35^5*x^4 +  56^6*x^5 + ...)*x^4/4

  + (1 + 5^2*x + 15^3*x^2 + 35^4*x^3 +  70^5*x^4 + 126^6*x^5 + ...)*x^5/5

  + (1 + 6^2*x + 21^3*x^2 + 56^4*x^3 + 126^5*x^4 + 252^6*x^5 + ...)*x^6/6

  + (1 + 7^2*x + 28^3*x^2 + 84^4*x^3 + 210^5*x^4 + 462^6*x^5 + ...)*x^7/7 + ...

MATHEMATICA

With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1, k]^(k+1)*x^(n+k)/n, {k, 0, m+2}], {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, n, binomial(m+k-1, k)^(k+1)*x^k)*x^m/m)+x*O(x^n)), n)}

m:=30;

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

Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1, k)^(k+1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021

(Sage)

m=30;

def A181066_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( exp( sum( sum( binomial(n+k-1, k)^(k+1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()

A181066_list(m) # G. C. Greubel, Apr 05 2021

CROSSREFS

Variants: A181070, A181076, A181078, A181080.

Cf. A181075 (log).

Sequence in context: A257030 A062495 A158889 * A078125 A034692 A002507

Adjacent sequences:  A181071 A181072 A181073 * A181075 A181076 A181077

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 02 2010

STATUS

approved

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Last modified May 6 09:45 EDT 2021. Contains 343580 sequences. (Running on oeis4.)