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A136658 Row sums of unsigned triangle A136656 and also of triangle 2*A136657.

%I A136658 #34 Jul 28 2018 07:23:45
%S A136658 1,2,10,68,580,5912,69784,933200,13912336,228390560,4088594464,
%T A136658 79186453568,1648396356160,36678170613632,868239454798720,
%U A136658 21776352497954048,576629116655862016,16069766602389885440,470015788927133039104,14392014594072635786240
%N A136658 Row sums of unsigned triangle A136656 and also of triangle 2*A136657.
%H A136658 Alois P. Heinz, <a href="/A136658/b136658.txt">Table of n, a(n) for n = 0..432</a>
%F A136658 a(n) = Sum_{k=0..n} (-1)^n*A136656(n,k), n>=0.
%F A136658 E.g.f.: exp(x*(2-x)/(1-x)^2) (from Jabotinsky type triangle).
%F A136658 a(n) = Sum_{k=0..n} Stirling1(n, k) * Bell(k) * (-1)^(n-k) * 2^k. - _Paul D. Hanna_, Dec 25 2011
%F A136658 a(n) = (3*n-1)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3). - _Vaclav Kotesovec_, Sep 25 2013
%F A136658 a(n) ~ 2^(1/6)*n^(n-1/6) * exp((n/2)^(1/3)+3*(n/2)^(2/3)-n-2/3) / sqrt(3) * (1 + 7/(27*(n/2)^(1/3)) - 422/(3645*(n/2)^(2/3))). - _Vaclav Kotesovec_, Sep 25 2013
%F A136658 Representation as special values of hypergeometric functions 2F2, in Maple notation: a(n) = (n+1)!*hypergeom([(1/2)*n+1, (1/2)*n+3/2], [3/2, 2], 1)*exp(-1), n = 1,2,... . - _Karol A. Penson_, Jul 28 2018
%p A136658 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A136658       binomial(n-1, j-1)*(j+1)!*a(n-j), j=1..n))
%p A136658     end:
%p A136658 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 01 2017
%t A136658 a[n_] := Sum[ StirlingS1[n, k] * BellB[k] * (-1)^(n-k) * 2^k, {k, 0, n}]; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Jul 09 2013, after _Paul D. Hanna_ *)
%t A136658 Table[Sum[BellY[n, k, (Range[n] + 1)!], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o A136658 (PARI)
%o A136658 {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A136658 {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
%o A136658 {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) * (-1)^(n-k)*2^k)}
%o A136658 /* _Paul D. Hanna_, Dec 25 2011 */
%Y A136658 Cf. A049376, A136656, A136657.
%K A136658 nonn,easy
%O A136658 0,2
%A A136658 _Wolfdieter Lang_, Feb 22 2008