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A181077 a(n) = Sum_{k=0..n-1} C(n-1,k)^(n-k) * n/(n-k). 4
1, 3, 10, 59, 726, 20832, 1405566, 202357171, 66675848266, 52415395776938, 88820554918533846, 339849475991699902472, 3175234567292428864024192, 65420235446121559438182151848, 2970041251569931717805628420162750 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
L.g.f.: L(x) = Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^n *x^k ] *x^n/n.
Logarithmic derivative of A181076.
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 59*x^4/4 + 726*x^5/5 +...
which equals the series:
L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
+ (1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 5^2*x^4 + 6^2*x^5 + ...)*x^2/2
+ (1 + 3^3*x + 6^3*x^2 + 10^3*x^3 + 15^3*x^4 + 21^3*x^5 + ...)*x^3/3
+ (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 + 35^4*x^4 + 56^4*x^5 + ...)*x^4/4
+ (1 + 5^5*x + 15^5*x^2 + 35^5*x^3 + 70^5*x^4 + 126^5*x^5 + ...)*x^5/5
+ (1 + 6^6*x + 21^6*x^2 + 56^6*x^3 + 126^6*x^4 + 252^6*x^5 + ...)*x^6/6
+ (1 + 7^7*x + 28^7*x^2 + 84^7*x^3 + 210^7*x^4 + 462^7*x^5 + ...)*x^7/7 + ...
Exponentiation yields the g.f. of A181076:
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 20*x^4 + 168*x^5 + 3659*x^6 + ... + A181076(n)*x^n + ...
MATHEMATICA
Table[Sum[Binomial[n-1, k]^(n-k)*n/(n-k), {k, 0, n-1}], {n, 25}] (* G. C. Greubel, Apr 05 2021 *)
PROG
(PARI) {a(n)=sum(k=0, n-1, binomial(n-1, k)^(n-k)*n/(n-k))}
(PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1, k)^m*x^k)*x^m/m)+x*O(x^n), n)}
(Magma) [(&+[Binomial(n-1, k)^(n-k)*n/(n-k): k in [0..n-1]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021
(Sage) [sum(binomial(n-1, k)^(n-k)*n/(n-k) for k in (0..n-1)) for n in (1..25)] # G. C. Greubel, Apr 05 2021
CROSSREFS
Cf. A181076 (exp), variants: A181073, A181075, A181079.
Sequence in context: A242953 A112101 A159321 * A158873 A103591 A245312
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 02 2010
STATUS
approved

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Last modified July 15 22:54 EDT 2024. Contains 374334 sequences. (Running on oeis4.)