A141056 -id:A141056 - cp's OEIS frontend

A141321 a(n) = -A141055(n)/(n+1)!.

1, 1, 10, 5, 42, 14, 60, 15, 110, 22, 5460, 910, 420, 60, 2040, 255, 11970, 1330, 23100, 2310, 15180, 1380, 163800, 13650, 3276, 252, 8120, 580, 286440, 19096, 314160, 19635, 3570, 210, 11515140, 639730, 103740, 5460

Offset: 0

Paul Curtz, Aug 02 2008

a(n+1)/a(n)= 2/2, 30/3, 2/4, 42/5, 2/6, 30/7, 2/8, 66/9, 2/10, 2730/11, 2/12 = A027760(n+2)/(n+1), see A141410. Numerators are also A141056(n+3).

Cf. A027760, A141055.

A141055 := proc(n) if n = 0 then -1; else procname(n-1)*A027760(n+2) ; end if; end proc:
A141321 := proc(n) -A141055(n)/(n+1)! ; end proc: # R. J. Mathar, Jul 08 2011

(* b = A141055 *) b[n_] := b[n] = b[n-1]*If[OddQ[n], 2, Denominator[BernoulliB[n+2]]]; b[0]=-1; a[n_] := -b[n]/(n+1)!; Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Dec 18 2014 *)

a(n)=if(n, my(pr=a(n-1)); fordiv(n+2, d, if(isprime(d+1), pr*=d+1)); pr, 1)/(n+1) \\ Charles R Greathouse IV, Jul 08 2011

a(2n) / a(2n+1) = n + 1.

A174129 Numerators of the first column of the table of fractions generated by the Akiyama-Tanigawa transform from a first row A164555(k)/A027642(k).

Original entry on oeis.org

1, 1, -1, -1, 31, 7, -1051, -201, 56911, 18311, -24346415, -4227881, 425739604981, 2082738855, -759610463437, -1935668684041, 91825384886337407, 3104887811293639, -333936446105326262497, -8039608511660213481, 496858217433153341005061

Offset: 0

Paul Curtz, Mar 09 2010

The first 6 rows if the table generated by iterative application of the Akiyama-Tanigawa transform starting with a header row of fractions  A164555(k)/A027642(k) are:
  1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, ...
  1/2, 2/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33, 5/6, 1382/455, ...
  -1/6, 1/3, 11/10, 6/5, -5/42, -13/7, -7/10, 68/15, 453/110, -175/11, ...
  -1/2, -23/15, -3/10, 554/105, 365/42, -243/35, -1099/30, 548/165, 19827/110, ...
  31/30, -37/15, -1171/70, -478/35, 469/6, 1247/7, -6153/22, -46708/33, ...
  7/2, 599/21, -129/14, -38566/105, -20995/42, 211515/77, 524699/66, ...
The numerators of the leftmost column define the current sequence.

D. Merlini, R. Sprugnoli and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A141056 (denominators), A174110, A174111 (first row).

Cf. A164555, A027642, A048993.

read("transforms3") ;
A174129 := proc(n) Lin := [bernoulli(0),-bernoulli(1),seq(bernoulli(k),k=2..n+1)] ; for r from 1 to n do Lin := AKIYATANI(Lin) ; end do; numer(op(1,Lin)) ; end proc:

a[0, k_] := a[0, k] = BernoulliB[k]; a[0, 1] = 1/2; a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); a[n_] := a[n, 0] // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 14 2012 *)

a(n) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)*B(j)), where B are the Bernoulli numbers A164555/A027642. - Fabián Pereyra, Jan 06 2022

A226156 a(n) = BS(n) * W(n) where BS = Sum_{k=0..n} ((-1)^k*k!/(k+1)) S(n, k) and S(n, k) the Stirling subset numbers A048993(n, k). W(n) = Product_{ p primes <= n+1 such that p divides n+1 or p-1 divides n } = A225481(n).

Original entry on oeis.org

1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0, -26315271553053477373, 0, 38089920879940267

Offset: 0

Peter Luschny, May 30 2013

a(n)/A225481(n) is a representation of the Bernoulli numbers. This is case m = 1 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 generalized Stirling subset numbers. A225481(n) can be seen as an analog of the Clausen numbers A141056(n). Reduced to lowest terms a(n)/A225481(n) becomes A027641(n)/A027642(n).

			The numerators of 1/1, -1/2, 1/6, 0/2, -1/30, 0/6, 1/42, 0/2, -1/30, 0/10, 5/66, 0/6, -691/2730, 0/14, 35/30, 0/2, -3617/510, 0/6, 43867/798, ... (the denominators are A225481(n)).

Peter Luschny, Stirling-Frobenius numbers
Peter Luschny, Generalized Bernoulli numbers.

Cf. A225481, A226157.

BS[n_] := Sum[((-1)^k*k!/(k + 1)) StirlingS2[n, k], {k, 0, n}];
W[n_] := Product[If[Divisible[n + 1, p] || Divisible[n, p - 1], p, 1], {p, Prime /@ Range[PrimePi[n + 1]]}];
a[n_] := BS[n] W[n];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jul 08 2019 *)

@CachedFunction
def EulerianNumber(n, k, m) :   # -- The Eulerian numbers --
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) + \
           (m*k+1)*EulerianNumber(n-1, k, m)
@CachedFunction
def SF_BS(n, m):   # -- The scaled Stirling-Frobenius Bernoulli numbers --
    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))
def A226156(n):    # -- The numerators of SF_BS(n, 1) relative to A225481 --
    C = mul(filter(lambda p: ((n+1)%p == 0) or (n%(p-1) == 0), primes(n+2)))
    return C*SF_BS(n, 1)
[A226156(n) for n in (0..25)]

A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

Original entry on oeis.org

1, 2, 6, 24, 30, 1440, 42, 120960, 30, 7257600, 66, 958003200, 2730, 5230697472000, 6, 62768369664000, 510, 64023737057280000

Offset: 0

Paul Curtz, Sep 28 2009

b(n)=a(2n+1)/a(2n) =2,4,48,2880,241920,145152,= 2*(1,2,24,1440,=1,2*A141421). Among other denominators, A027642,A141056,A164020. 2*A141421 is second bisection of A091137 which is linked to Bernoulli via A027760. See A160014,von Staudt-Clausen theorem.

A225480 a(n) = B2(n) * C(n) where B2(n) are generalized Bernoulli numbers and C(n) the Clausen numbers.

Original entry on oeis.org

1, 0, -2, 0, 14, 0, -62, 0, 254, 0, -5110, 0, 2828954, 0, -114674, 0, 237036478, 0, -11499383114, 0, 183092554714, 0, -3584085584926, 0, 3965530936622474, 0, -573989008898786, 0, 6375197353574922166, 0, -9251189109760413581110, 0, 33111281730973040956798, 0

Offset: 0

Peter Luschny, May 30 2013

The Clausen numbers C(n) are T(n, 1) in A160014.

			The numerators of 1/1, 0/2, -2/6, 0/2, 14/30, 0/2, -62/42, 0/2, 254/30, 0/2, -5110/66, 0/2, 2828954/2730, ... (the denominators are the Clausen numbers).

Peter Luschny, Stirling-Frobenius numbers
Peter Luschny, Generalized Bernoulli numbers.

Cf. A141056, A160014.

B := (n, m) -> add(add(add(((-1)^(n-v)/(j+1))*binomial(n,k)*binomial(j, v)*(m*v)^k, v = 0..j), j = 0..k), k = 0..n);
C := proc(n) numtheory[divisors](n);map(i->i+1,%);select(isprime,%);mul(i,i=%) end:
A225480 := n -> B(n, 2)*C(n); seq(A225480(n), n = 0..33);

B[n_, m_] := Sum[((-1)^(n - v)/(j + 1))*Binomial[n, k]*Binomial[j, v]*If[k == 0, 1, (m*v)^k], {k, 0, n}, {j, 0, k}, {v, 0, j}];
c[n_] := Denominator[Sum[Boole[PrimeQ[d + 1]]/(d + 1), {d, Divisors[n]}]];
a[n_] := B[n, 2]*c[n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Aug 02 2019, from Maple *)

@CachedFunction
def EulerianNumber(n, k, m) :   # The Eulerian numbers
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1,k-1,m)+(m*k+1)*EulerianNumber(n-1,k,m)
@CachedFunction
def B(n, m):   # The generalized Bernoulli numbers
    return add(add(EulerianNumber(n, j, m)*binomial(j, n - k)
           for j in (0..n))*(-1)^k/(k+1) for k in (0..n))
def A225480(n):
    if n == 0: return 1
    C = mul(filter(lambda s: is_prime(s) , map(lambda i: i+1, divisors(n))))
    return C*B(n, 2)
print([A225480(n) for n in (0..33)])

Let B(n,m) = sum_{k = 0..n} sum_{j = 0..k} sum_{v = 0..j} ((-1)^(n-v)/(j+1)) *binomial(n,k)*binomial(j,v)*(m*v)^k then a(n) = B(n,2)*A141056(n).

Let B2(n) = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) where S_{2}(n, k) the Stirling-Frobenius subset numbers A039755(n, k) then a(n) = B2(n)*A141056(n).

A229023 Numerators of the main diagonal of A225825 difference table, a sequence linked to Bernoulli, Genocchi and Clausen numbers.

Original entry on oeis.org

1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272, -90228445612544, 19078660567134208, -2034677178643867648, 123160010212358914048, -19182197131374977024, 228111332170536254898176, -51166426240975948419354886144

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 11 2013

a(n) is divisible by 2^n and congruent to 1, 2, 4, 5, 7 or 8 mod 9.

			1, -2/3, 16/15, -424/105, 2944/105, -70240/231, 70873856/15015, ...

Cf. A181131 (denominators), A225825, A110501 (Genocchi numbers), A141056 (Clausen numbers), A212196 (Bernoulli medians), A005439 (Genocchi medians).

nmax = 30; Clausen[n_] := Times @@ Select[Divisors[n] + 1, PrimeQ]; t = Join[{1}, Table[Numerator[BernoulliB[n, 1/2] - (n + 1)*EulerE[n, 0]]/Clausen[n], {n, 1, nmax}]]; dt = Table[Differences[t, n], {n, 0, nmax}]; Diagonal[dt] // Numerator

A249306 Denominators A027642(n) of Bernoulli numbers except for a(4*k+5)=2 instead of 1.

Original entry on oeis.org

1, 2, 6, 1, 30, 2, 42, 1, 30, 2, 66, 1, 2730, 2, 6, 1, 510, 2, 798, 1, 330, 2, 138, 1, 2730, 2, 6, 1, 870, 2, 14322, 1, 510, 2, 6, 1, 1919190, 2, 6, 1, 13530, 2, 1806, 1, 690, 2, 282, 1, 46410, 2, 66, 1, 1590, 2, 798, 1, 870, 2, 354, 1

Offset: 0

Paul Curtz, Oct 28 2014

nonn

There exist an infinity of 1's, 2's, 6's, 30's, 42's, 66's, ... .
Respective ranks:
   0,   3,    7,   11,   15,   19, ...
   1,   5,    9,   13,   17,   21, ...   (= A016813)
   2,  14,   26,   34,   38,   62, ...   (= A051222)
   4,   8,   68,   76,  124,  152, ...   (= A051226)
   6, 114,  186,  258,  354,  402, ...   (= A051228)
  10,  50,  170,  370,  470,  590, ...   (= A051230)
  12,  24, 1308, 1884, 2004, 2364, ...   (= A249134)
  etc.
Hence by antidiagonals a permutation of A001477(n).
First column: A248614(n).
a(n) is an alternative sequence for the denominators of the Bernoulli numbers.
First 36 terms of the corresponding clockwise spiral:
.
  330------2----138------1---2730------2
    |                                  |
    |                                  |
    1     42------1-----30------2      6
    |      |                    |      |
    |      |                    |      |
  798      2      1------2     66      1
    |      |             |      |      |
    |      |             |      |      |
    2     30------1------6      1    870
    |                           |      |
    |                           |      |
  510------1------6------2---2730      2
                                       |
                                       |
    1------6------2----510------1--14322

Index entries for sequences related to Bernoulli numbers

A variant of the Clausen numbers A141056, A160014. And of A176591.

Cf. A000034, A002445, A016813, A027642, A051222, A051226, A051228, A051230, A090126, A164020, A248614, A249134.

Clausen := proc(n) local S, i;
S := numtheory[divisors](n); S := map(i->i+1, S);
S := select(isprime, S); mul(i, i=S) end:
A249306 := n -> `if`(n mod 4 = 3, 1, Clausen(n)):
seq(A249306(n), n=0..59); # Peter Luschny, Nov 10 2014

a[n_] := Denominator[BernoulliB[n]]; a[n_ /; Mod[n, 4] == 1] = 2; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2014 *)

a(2n) = A002445(n), a(2n+1) = A000034(n+1).

A368117 a(n) = Product_{(s - 2)|n, s prime} s if n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 3, 15, 3, 21, 15, 3, 3, 165, 21, 39, 15, 3, 3, 1785, 3, 57, 165, 3, 21, 345, 39, 3, 15, 21, 3, 4785, 3, 93, 1785, 3, 3, 195, 57, 777, 165, 3, 3, 615, 21, 129, 345, 3, 39, 922845, 3, 3, 15, 3, 21, 15105, 3, 3, 4785, 273, 3, 885, 93, 183, 1785, 3, 3

Offset: 0

Peter Luschny, Dec 13 2023

nonn

All terms are squarefree.

			The divisors of 15 are {1, 3, 5, 15}. Adding 2 to the divisors gives {3, 5, 7, 17}, which are all primes. Therefore a(15) = 3*5*7*17 = 1785.

Cf. A160014, A007947 (radical of n, m=0), A141056 and A027760 (Clausen numbers, m=1).

{1}~Join~Array[Times @@ Select[Divisors[#] + 2, PrimeQ] &, 62] (* Michael De Vlieger, Dec 14 2023 *)

a(n) = if (n>0, my(d=divisors(n)); prod(k=1, #d, if (isprime(p=d[k]+2), p, 1)), 1); \\ Michel Marcus, Dec 15 2023

def a(n): return (mul(s for s in map(lambda i: i + 2, divisors(n))
                  if is_prime(s)) if n > 0 else 1)
print([a(n) for n in range(63)])

a(n) = A160014(2, n).

A160035 Clausen-normalized numerators of the Bernoulli numbers of order 2.

Original entry on oeis.org

1, 0, -1, 0, 3, 0, -5, 0, 7, 0, -45, 0, 7601, 0, -91, 0, 54255, 0, -745739, 0, 3317609, 0, -17944773, 0, 5436374093, 0, -213827575, 0, 641235447783, 0, -249859397004145, 0, 238988952277727, 0, -85063699326111, 0, 921034504356871708055, 0, -108409774812137683

Offset: 0

Peter Luschny, Apr 30 2009

sign

Let B_n{^(k)}(x) denote the Bernoulli polynomials of order k, defined by the generating function
(t/(exp(t)-1))^k*exp(x*t) = Sum_{n>=0} B_n{^(k)}(x) t^n/n!
Bernoulli numbers of order 1 (defined as B_n{^(1)}(1)) can be regarded as a pair of sequences B1_n = N1_n / D1_n with
N1_n = A027641, D1_n = A141056 (Clausen).
Similarly Bernoulli numbers of order 2 (defined as B_n{^(2)}(1)) can be regarded as a pair of sequences B2_n = N2_n / D2_n with
N2_n = this sequence, D2_n = A141056 (Clausen).

			The Clausen-normalized Bernoulli polynomials of order 2 are:
1
2 x - 2
6 x^2 - 12 x + 5
2 x^3 - 6 x^2 + 5 x - 1
30 x^4 - 120 x^3 + 150 x^2 - 60 x + 3
2 x^5 - 10 x^4 + 50/3 x^3 - 10 x^2 + x + 1/3
42 x^6 - 252 x^5 + 525 x^4 - 420 x^3 + 63 x^2 + 42 x - 5
The value of these polynomials at x = 1 gives the sequence.

L. Comtet, Advanced Combinatorics, Reidel, Boston, Mass., 1974.
C. Jordan, Calculus of Finite Differences, New York, Chelsea, 1965.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Berlin, Springer-Verlag, 1924.

Cf. A120282, A132094, A100615 and A027643.

aList := proc(n) local g,c,i; g := k -> (t/(exp(t)-1))^k*exp(x*t): c := proc(n) local i; mul(i,i=select(isprime, map(i->i+1,numtheory[divisors](n)))) end: convert(series(g(2),t,n+8),polynom): seq(i!*c(i)*subs(x=1,coeff(%,t,i)),i=0..n) end: aList(38);

aList[n_] := Module[{g, c, s},
   g[k_] := (t/(Exp[t]-1))^k*Exp[x*t];
   c[k_] := Times @@ Select[Divisors[k]+1, PrimeQ];
   s = Series[g[2], {t, 0, n + 8}] // Normal;
Join[{1}, Table[i!*c[i]*Coefficient[s, t, i] /. x -> 1, {i, 1, n}]]];
aList[38] (* Jean-François Alcover, May 28 2024, after Peter Luschny *)

A174182 Numerator of the first column, n-th row of the table of the Akiyama-Tanigawa transform starting from a top row of Bernoulli numbers.

Original entry on oeis.org

1, 3, 17, 13, 481, 69, 1595, 53, 64561, 19333, -24278897, -4223787, 425750784331, 2082755237, -759610365139, -1935668618507, 91825384919760257, 3104887811555781, -333936446105117072383, -8039608511659164907, 496858217433153687034811, 31900258438443561908965, -1108179772136293880993162549, -186044136772398390757763787, 167280081459577193334628789960171

Offset: 0

Paul Curtz, Mar 11 2010

Starting with a top row of Bernoulli numbers, the Akiyama-Tanigawa transform generates further rows as follows:
  1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66,...
  3/2, -4/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33,..
  17/6, -11/3, 11/10, 6/5, -5/42, -13/7, -7/10, 68/15, 453/110,...
  13/2, -143/15, -3/10, 554/105, 365/42, -243/35, -1099/30, 548/165,...
  481/30, -277/15, -1171/70, -478/35, 469/6, 1247/7, -6153/22,..
  69/2, -73/21, -129/14, -38566/105, -20995/42, 211515/77,...
The numerators of the leftmost column define the current sequence.
The denominators appear to be the same as A141056.

D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A174129, A141056, A051049.

a[0, k_] := BernoulliB[k]; a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); Table[a[n, 0], {n, 0, 24}] // Numerator (* Jean-François Alcover, Sep 18 2012 *)

(a(n)-A174129(n))/A141056(n) = A000225(n).

cp's OEIS Frontend

A141321 a(n) = -A141055(n)/(n+1)!.

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A174129 Numerators of the first column of the table of fractions generated by the Akiyama-Tanigawa transform from a first row A164555(k)/A027642(k).

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A226156 a(n) = BS(n) * W(n) where BS = Sum_{k=0..n} ((-1)^k*k!/(k+1)) S(n, k) and S(n, k) the Stirling subset numbers A048993(n, k). W(n) = Product_{ p primes <= n+1 such that p divides n+1 or p-1 divides n } = A225481(n).

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A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

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A225480 a(n) = B2(n) * C(n) where B2(n) are generalized Bernoulli numbers and C(n) the Clausen numbers.

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A229023 Numerators of the main diagonal of A225825 difference table, a sequence linked to Bernoulli, Genocchi and Clausen numbers.

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A249306 Denominators A027642(n) of Bernoulli numbers except for a(4*k+5)=2 instead of 1.

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A368117 a(n) = Product_{(s - 2)|n, s prime} s if n > 0, a(0) = 1.

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