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.

-1, 1, 1, 1, 19, 27, 863, 1375, 33953, 57281, 3250433, 5675265, 13695779093, 24466579093, 132282840127, 240208245823, 111956703448001, 205804074290625, 151711881512390095, 281550972898020815, 86560056264289860203, 161867055619224199787, 20953816286242674495191, 39427936010479474495191

Offset: 0

Author

Paul Curtz, Aug 05 2008

Keywords

• sign

Comments

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

-1, 1, 1, 1, 19, 27, 863, ...

1, -3, 5, 1, 11, 11, 271, ...

-1, 5, -23, 9, 19, 11, 191, ...

1, -7, 53, -55, 251, 27, 271, ...

-1, 9, -95, 161, -1901, 475, 863, ...

1, -11, 149, -351, 6731, -4277, 19087, ...

...

The first two rows are related via T(0,j) = A027760(j)*T(0,j-1) - T(1,j).

References

• P. Curtz, Integration .., note 12, C.C.S.A., Arcueil, 1969.

Crossrefs

Cf. A141047, A140811, A140825.

Programs

Maple

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:
A048994 := proc(n, k) combinat[stirling1](n, k) ; end proc:
a := proc(i,j) add(A048994(j,k)*x^k,k=0..j) ; int(%,x=i..i+1) ; %/j! ; end proc:
A141417 := proc(n) (-1)^(n+1)*A091137(n)*a(0,n) ; end proc:
seq(A141417(n),n=0..40) ; # R. J. Mathar, Nov 17 2010

Mathematica

(* 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 *)

Maxima

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 */

Formula

a(i,j) = a(i-1,j) + a(i-1,j-1), see reference page 33.

(q+1-j)*Sum_{j=0..q} a(i,j)*(-1)^(q-j) = binomial(i,q), see reference page 35.

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

Extensions

Erroneous formula linking A091137 and A002196 removed, and more terms and program added by R. J. Mathar, Nov 17 2010