The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A226157 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A226157

a(n) = BS2(n) * W(n) where BS2 = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) and S_{2}(n, k) are the Stirling-Frobenius subset numbers A039755(n, k). W(n) = product{p primes <= n+1 such that p divides n+1 or p-1 divides n} = A225481(n).
(history; published version)

#20 by Peter Luschny at Thu Jun 27 08:40:20 EDT 2019

STATUS

editing

approved

#19 by Peter Luschny at Thu Jun 27 08:40:04 EDT 2019

COMMENTS

a(n)/A225481(n) is case m = 2 of the scaled generalized Bernoulli numbers defined as sum_Sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{m}(n, k) where S_{m}(n, k) are Stirling-Frobenius subset numbers. A225481(n) can be seen as an analog of the Clausen numbers A141056(n).

PROG

return ((m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) + \

(m*k+1)*EulerianNumber(n-1, k, m))

return (add(add(EulerianNumber(n, j, m)*binomial(j, n - k) \

for j in (0..n))/((-m)^k*(k+1)) for k in (0..n)))

STATUS

proposed

editing

Discussion

Thu Jun 27

08:40

Peter Luschny: Thanks!

#18 by Jean-François Alcover at Thu Jun 27 08:12:34 EDT 2019

STATUS

editing

proposed

#17 by Jean-François Alcover at Thu Jun 27 08:12:30 EDT 2019

MATHEMATICA

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = If[n == 0, If[k == 0, 1 , 0], (m*(n-k) + m - 1)*EulerianNumber[n-1, k-1, m] + (m*k + 1)* EulerianNumber[n-1, k, m]];

BS[n_, m_] := Sum[Sum[EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/ ((-m)^k*(k+1)), {k, 0, n}]

a[n_] := Product[If[Divisible[n+1, p] || Divisible[n, p-1], p, 1], {p, Prime /@ Range @ PrimePi[n+1]}] * BS[n, 2];

Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 27 2019, from Sage *)

STATUS

approved

editing

#16 by Charles R Greathouse IV at Thu Sep 26 01:20:01 EDT 2013

STATUS

proposed

approved

#15 by Michel Marcus at Wed Sep 25 11:38:23 EDT 2013

STATUS

editing

proposed

#14 by Michel Marcus at Wed Sep 25 11:38:19 EDT 2013

COMMENTS

a(n)/A225481(n) is case m = 2 of the scaled generalized Bernoulli numbers defined as sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{m}(n, k) where S_{m}(n, k) are Stirling-Frobenius subset numbers. A225481(n) can be seen as an analogon analog of the Clausen numbers A141056(n).

STATUS

approved

editing

#13 by Bruno Berselli at Fri Jun 07 17:48:44 EDT 2013

STATUS

reviewed

approved

#12 by Joerg Arndt at Fri Jun 07 11:27:23 EDT 2013

STATUS

proposed

reviewed

#11 by Peter Luschny at Fri Jun 07 07:42:56 EDT 2013

STATUS

editing

proposed