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Row sums of unsigned triangle A075513.
3

%I #41 Oct 09 2018 15:13:53

%S 1,3,18,170,2200,36232,725200,17095248,463936896,14246942336,

%T 488428297984,18491942300416,766293946203136,34498781924766720,

%U 1676731077272217600,87501958444207351808,4880017252828686155776

%N Row sums of unsigned triangle A075513.

%H Vincenzo Librandi, <a href="/A074932/b074932.txt">Table of n, a(n) for n = 1..200</a>

%H Victor J. W. Guo, Yiting Yang, <a href="https://arxiv.org/abs/1704.01295">Proof of a conjecture of Kløve on permutation codes under the Chebychev distance</a>, arXiv:1704.01295 [cs.IT], 2017. Also in Designs, Codes and Cryptography, June 2017, Volume 83, Issue 3, pp 685-690.

%H Torleiv Kløve, <a href="http://www.ii.uib.no/publikasjoner/texrap/pdf/2008-376.pdf"> Spheres of Permutations under the Infinity Norm - Permutations with limited displacement</a>, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

%F a(n) = sum(|A075513(n, m)|, m=0..n-1) = sum(binomial(n-1, m)*(m+1)^(n-1), m=0..n-1), n>=1.

%F E.g.f.: log(G(x)) where G(x) = (1 + exp(2*x*G(x)))/2 is the e.g.f. of A201595. - _Paul D. Hanna_, Aug 02 2012

%F E.g.f: log(x-LambertW(-x*exp(x)))-log(2*x). - _Vaclav Kotesovec_, Dec 04 2012

%F a(n) ~ n!/(sqrt(2*Pi*(1+LambertW(exp(-1))))*n^(3/2)*LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Dec 04 2012

%F a(n) = A072034(n)/n. - _Vladimir Reshetnikov_, Nov 09 2016

%F O.g.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - k*x)^k. - _Ilya Gutkovskiy_, Oct 09 2018

%e E.g.f.: A(x) = x + 3*x^2/2! + 18*x^3/3! + 170*x^4/4! + 2200*x^5/5! +...

%e where exp(A(x)) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! + 67328*x^6/6! +...+ A201595(n)*x^n/n! +...

%t Rest[CoefficientList[Series[Log[x-LambertW[-x*Exp[x]]]-Log[2*x], {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Dec 04 2012 *)

%t a[n_] := Sum[Binomial[n-1, k]*(k+1)^(n-1), {k, 0, n-1}]; Table[a[n], {n, 1, 17}] (* _Jean-François Alcover_, Jul 09 2013, after _Paul D. Hanna_ *)

%o (PARI) {a(n)=sum(k=0,n-1,binomial(n-1,k)*(k+1)^(n-1))} \\ _Paul D. Hanna_, Aug 02 2012

%o (PARI) {a(n)=local(A201595=serreverse(x-x*tanh(x+x^2*O(x^n)))/x);n!*polcoeff(log(A201595), n)} \\ _Paul D. Hanna_, Aug 02 2012

%o (PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for(k=1, n, A = log( (1 + exp( 2*x * exp(A))) / 2 )); n! * polcoeff(A, n))}; /* _Michael Somos_, Apr 10 2018 */

%Y Cf. A201595.

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Oct 02 2002