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A141417 (-1)^(n+1)*A091137(n)*a(0,n), where a(i,j) = Integral_{x=i..i+1} x*(x-1)*(x-2)*...*(x-j+1)/j! dx.

%I A141417 #19 Aug 09 2022 00:15:33
%S A141417 -1,1,1,1,19,27,863,1375,33953,57281,3250433,5675265,13695779093,
%T A141417 24466579093,132282840127,240208245823,111956703448001,
%U A141417 205804074290625,151711881512390095,281550972898020815,86560056264289860203,161867055619224199787,20953816286242674495191,39427936010479474495191
%N A141417 (-1)^(n+1)*A091137(n)*a(0,n), where a(i,j) = Integral_{x=i..i+1} x*(x-1)*(x-2)*...*(x-j+1)/j! dx.
%C A141417 This is row i=0 of an array defined as T(i,j) = (-1)^(i+j+1)*A091137(j)*a(i,j), columns j >= 0, which starts
%C A141417   -1,   1,   1,    1,    19,    27,   863, ...
%C A141417    1,  -3,   5,    1,    11,    11,   271, ...
%C A141417   -1,   5, -23,    9,    19,    11,   191, ...
%C A141417    1,  -7,  53,  -55,   251,    27,   271, ...
%C A141417   -1,   9, -95,  161, -1901,   475,   863, ...
%C A141417    1, -11, 149, -351,  6731, -4277, 19087, ...
%C A141417   ...
%C A141417 The first two rows are related via T(0,j) = A027760(j)*T(0,j-1) - T(1,j).
%D A141417 P. Curtz, Integration .., note 12, C.C.S.A., Arcueil, 1969.
%F A141417 a(i,j) = a(i-1,j) + a(i-1,j-1), see reference page 33.
%F A141417 (q+1-j)*Sum_{j=0..q} a(i,j)*(-1)^(q-j) = binomial(i,q), see reference page 35.
%F A141417 a(n) = numerator(n*(n+1)*Sum_{k=1..n} ((-1)^(n-k)*Stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1))), n>0, a(0)=-1. - _Vladimir Kruchinin_, Dec 12 2016
%p A141417 A091137 := proc(n) local a, i, p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end proc:
%p A141417 A048994 := proc(n, k) combinat[stirling1](n, k) ; end proc:
%p A141417 a := proc(i,j) add(A048994(j,k)*x^k,k=0..j) ; int(%,x=i..i+1) ; %/j! ; end proc:
%p A141417 A141417 := proc(n) (-1)^(n+1)*A091137(n)*a(0,n) ; end proc:
%p A141417 seq(A141417(n),n=0..40) ; # _R. J. Mathar_, Nov 17 2010
%t A141417 (* a7 = A091137 *) a7[n_] := a7[n] = Times @@ Select[ Divisors[n]+1, PrimeQ]*a7[n-1]; a7[0]=1; a[n_] := (-1)^(n+1) * a7[n] * Integrate[ (-1)^n*Pochhammer[-x, n], {x, 0, 1}]/n!; Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Aug 10 2012 *)
%o A141417 (Maxima)
%o A141417 a(n):=if n=0 then -1 else num(n*(n+1)*sum(((-1)^(n-k)*stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1)),k,1,n)); /* _Vladimir Kruchinin_, Dec 12 2016 */
%Y A141417 Cf. A141047, A140811, A140825.
%K A141417 sign
%O A141417 0,5
%A A141417 _Paul Curtz_, Aug 05 2008
%E A141417 Erroneous formula linking A091137 and A002196 removed, and more terms and program added by _R. J. Mathar_, Nov 17 2010