cp's OEIS Frontend

A162509 Row sums of the absolute values of a triangular array related to the Bernoulli numbers.

%I A162509 #48 Apr 17 2018 17:22:53
%S A162509 1,1,4,20,124,932,8284,85220,997084,13082852,190320604,3040770020,
%T A162509 52937870044,997533561572,20228969244124,439283696014820,
%U A162509 10170742982007004,250110224694309092,6510327792455418844,178832105312143131620,5169772417850111583964
%N A162509 Row sums of the absolute values of a triangular array related to the Bernoulli numbers.
%C A162509 Let T(n,k) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1)^(n-1) for n >= 1, k >= 1 and additionally T(0,0) = 1. Then a(n) = sum_{k=0..n} abs(T(n,k)).
%C A162509 a(n) = A073146(n,n-1) for n >= 1.
%C A162509 a(n) appears to be the total number of subsets over all chains of the poset on the powerset of {1,2,...,n-1} ordered by set inclusion.  That is,  a(n) = Sum_{k=0..n} A038719(n,k)*(k+1).  For example a(2)=4 because there are three chains: {}; {1}; {},{1}; and there are 4 total subsets. - _Geoffrey Critzer_, Nov 28 2014
%H A162509 Bruno Berselli, <a href="/A162509/b162509.txt">Table of n, a(n) for n = 0..100</a>
%F A162509 a(n+1)=Sum_{k, 0<=k<=n} A199400(n,k) = Sum_{k, 0<=k<=n} A199335(n,k)*2^k. - _Philippe Deléham_, Nov 06 2011
%F A162509 G.f.: 1+x/(1-4x/(1-x/(1-6x/(1-2x/(1-8x/(1-3x/(1-10x/(1-4x/1-....)))))))) (continued fraction). - _Philippe Deléham_, Nov 22 2011
%F A162509 G.f.: 1 + x/Q(0), where Q(k) = 1 - x*(3*k+4) - 2*x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 03 2013
%F A162509 a(n + 1) = sum {k >= 0} (k*(k + 1)^n)/2^(k + 1) for n >= 0. Comparison with the formula A000670(n) = sum {k >= 0} (k^n)/2^(k + 1) yields a(n + 1) = sum {k = 0..n} binomial(n,k)*A000670(k + 1). - _Peter Bala_, Jul 21 2014
%F A162509 a(n) ~ n! / log(2)^(n+1). - _Vaclav Kotesovec_, Apr 17 2018
%p A162509 A162508 := proc(n,k) local v; if n=0 and k=0 then 1 else
%p A162509 add((-1)^v*v*binomial(k,v)*(v+1)^(n-1),v=0..k) fi end:
%p A162509 a := proc(n) local k; add(abs(A162508(n,k)),k=0..n) end:
%t A162509 t[0, 0] = 1; t[n_, k_] := Sum[(-1)^v*v*Binomial[k, v]*(v+1)^(n-1), {v, 0, k}]; a[n_] := Sum[Abs[t[n, k]], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jun 28 2013 *)
%o A162509 (Sage)
%o A162509 def A162509(n):
%o A162509     return add(abs(A162508(n, k)) for k in (0..n))
%o A162509 [A162509(n) for n in (0..20)] # _Peter Luschny_, Jul 21 2014
%Y A162509 Cf. A162508, A073146.
%K A162509 nonn
%O A162509 0,3
%A A162509 _Peter Luschny_, Jul 05 2009