A074932 Row sums of unsigned triangle A075513.
1, 3, 18, 170, 2200, 36232, 725200, 17095248, 463936896, 14246942336, 488428297984, 18491942300416, 766293946203136, 34498781924766720, 1676731077272217600, 87501958444207351808, 4880017252828686155776
Offset: 1
Examples
E.g.f.: A(x) = x + 3*x^2/2! + 18*x^3/3! + 170*x^4/4! + 2200*x^5/5! +... 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! +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- Victor J. W. Guo, Yiting Yang, Proof of a conjecture of Kløve on permutation codes under the Chebychev distance, arXiv:1704.01295 [cs.IT], 2017. Also in Designs, Codes and Cryptography, June 2017, Volume 83, Issue 3, pp 685-690.
- Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
Crossrefs
Cf. A201595.
Programs
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Mathematica
Rest[CoefficientList[Series[Log[x-LambertW[-x*Exp[x]]]-Log[2*x], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 04 2012 *) 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 *) -
PARI
{a(n)=sum(k=0,n-1,binomial(n-1,k)*(k+1)^(n-1))} \\ Paul D. Hanna, Aug 02 2012 -
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 -
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 */
Formula
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.
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
E.g.f: log(x-LambertW(-x*exp(x)))-log(2*x). - Vaclav Kotesovec, Dec 04 2012
a(n) ~ n!/(sqrt(2*Pi*(1+LambertW(exp(-1))))*n^(3/2)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Dec 04 2012
a(n) = A072034(n)/n. - Vladimir Reshetnikov, Nov 09 2016
O.g.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - k*x)^k. - Ilya Gutkovskiy, Oct 09 2018