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

0,5

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.

Links

Table of n, a(n) for n=0..23.

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

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 *)

Prog

(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 */

Crossrefs

Cf. A141047, A140811, A140825.

Sequence in context: A146651 A146808 A147232 * A264834 A069529 A138335

Adjacent sequences: A141414 A141415 A141416 * A141418 A141419 A141420

Keyword

sign

Author

Paul Curtz, Aug 05 2008

Extensions

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

Status

approved