A091137 -id:A091137 - cp's OEIS frontend

A232853 Repeat n+1 times A091137(n).

1, 2, 2, 12, 12, 12, 24, 24, 24, 24, 720, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 3628800

Offset: 0

Paul Curtz, Dec 01 2013

A002657(n) and A091137(n) are linked to the Bernoulli numbers B n.
Unreduced differences table of A002657(n)/A091137(n):
1,            1/2,     5/12,    9/24,  251/720, 475/1440,...
-1/2,       -1/12,    -1/24, -19/720, -27/1440,... =-A141417(n+1)/A091137(n+1),
5/12,        1/24,   11/720, 11/1440,...
-9/24,    -19/720, -11/1440,...
251/720,  27/1440,...
-475/1440,... etc.
This is an autosequence of the second kind: its inverse binomial transform is the signed sequence with the main diagonal double of the first upper diagonal.
a(n) is the denominators written by antidiagonals.

			1,
2,   2,
12, 12, 12,
24, 24, 24, 24, etc.

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Centre de Calcul Scientifique de l'Armement, Arcueil 1969. Pages 36 and 56.

Cf. A195287, A002208/A002209 (reduced autosequence), A193546, A174727, A165313.

Repeat n+1 times A091137(n). Triangle.

A176493 A091137(n)/(n+1).

Original entry on oeis.org

1, 1, 4, 6, 144, 240, 8640, 15120, 403200, 725760, 43545600, 79833600, 201180672000, 373621248000, 2092278988800, 3923023104000, 1883051089920000, 3556874280960000, 2688996956405760000, 5109094217170944000, 1605715325396582400000, 3065456530302566400000

Offset: 0

Paul Curtz, Apr 19 2010

nonn

a(n) = A174727(n)/A027760(n+1).

A195338 a(n) = A091137(n) / A016116(n).

Original entry on oeis.org

1, 2, 6, 12, 180, 360, 7560, 15120, 226800, 453600, 14968800, 29937600, 40864824000, 81729648000

Offset: 0

Paul Curtz, Sep 16 2011

nonn

a[n_] := (For[primePi = 1; p = 2; m = 1, p <= n + 1, primePi++, p = Prime[primePi]; m = m*p^Quotient[n, p - 1]]; m/2^Floor[n/2]); Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Oct 02 2012 *)

A027760 Denominator of Sum_{p prime, p-1 divides n} 1/p.

Original entry on oeis.org

2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2

Offset: 1

N. J. A. Sloane

The GCD of integers x^(n+1)-x, for all integers x. - Roger Cuculiere (cuculier(AT)imaginet.fr), Jan 19 2002
If each x in a ring satisfies x^(n+1)=x, the characteristic of the ring is a divisor of a(n) (Rosenblum 1977). - Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Sep 24 2008
The denominators of the Bernoulli numbers for n>0. B_n sequence begins 1, -1/2, 1/6, 0/2, -1/30, 0/2, 1/42, 0/2, ... This is an alternative version of A027642 suggested by the theorem of Clausen. To add a(0) = 1 has been proposed in A141056. - Peter Luschny, Apr 29 2009
For N > 1, a(n) is the greatest number k such that x*y^n ==y*x^n (mod k) for any integers x and y.  Example: a(19) = 798 because x*y^19 ==y*x^19 (mod 798). - Michel Lagneau, Apr 21 2012
a(n) is the largest k such that b^(n+1) == b (mod k) for every integer b. - Mateusz Szymański, Feb 18 2016, corrected by Thomas Ordowski, Jul 01 2018
When n is even, a(n) is the product of the distinct primes dividing the denominator of zeta(1-n), where zeta(s) is the Riemann zeta function. - Griffin N. Macris, Jun 13 2016
If n+1 is prime, then A002322(a(n)) = n. Composite numbers n+1 such that A002322(a(n)) = n are in A317210. - Max Alekseyev and Thomas Ordowski, Jul 09 2018

			1/2, 5/6, 1/2, 31/30, 1/2, 41/42, 1/2, 31/30, 1/2, 61/66, 1/2, 3421/2730, 1/2, 5/6, 1/2, 557/510, ...

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas Clausen, Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen, Astr. Nachr. 17 (1840), 351-352. [_Peter Luschny_, Apr 29 2009]
S. C. Locke and A. Mandel, Problem E 2901, American Mathematical Monthly 88 (1981), p. 538. Solution in Vol. 90 (1983), pp. 212-213. [Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Jul 31 2008]
D. M. Rosenblum, Problem 1019, Mathematics Magazine 50 (1977), p. 164. Solution by T. Orloff in Vol. 52 (1979), p. 50.
Wikipedia, Bernoulli number

Cf. A002997, A027759, A027642, A141056, A317210.

A027760 := proc(n) local s,p; s := 0 ; p := 2; while p <= n+1 do if n mod (p-1) = 0 then s := s+1/p; fi; p := nextprime(p) ; od: denom(s) ; end: # R. J. Mathar, Aug 12 2008

clausen[n_] := Product[i, {i, Select[ Map[ # + 1 &, Divisors[n]], PrimeQ]}]
Table[clausen[i], {i, 1, 20}] (* Peter Luschny, Apr 29 2009 *)
f[n_] := Times @@ Select[Divisors@n + 1, PrimeQ]; Array[f, 56] (* Robert G. Wilson v, Apr 25 2012 *)

a(n)=denominator(sumdiv(n,d,if(isprime(d+1),1/(d+1)))) \\ Charles R Greathouse IV, Jul 08 2011

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

def A027760(n):
    return mul(filter(lambda s: is_prime(s), map(lambda i: i+1, divisors(n))))
[A027760(n) for n in (1..56)]  # Peter Luschny, May 23 2013

a(2*k) = A091137(2*k)/A091137(2*k-1). - Paul Curtz, Aug 05 2008
a(n) = product_{p prime, p-1 divides n}. - Eric M. Schmidt, Aug 01 2013
a(2n-1) = 2. - Robert G. Wilson v, Jul 23 2018

Formula submitted with A141417 added by R. J. Mathar, Nov 17 2010

A140811 a(n) = 6*n^2 - 1.

Original entry on oeis.org

-1, 5, 23, 53, 95, 149, 215, 293, 383, 485, 599, 725, 863, 1013, 1175, 1349, 1535, 1733, 1943, 2165, 2399, 2645, 2903, 3173, 3455, 3749, 4055, 4373, 4703, 5045, 5399, 5765, 6143, 6533, 6935, 7349, 7775, 8213, 8663, 9125, 9599, 10085, 10583, 11093, 11615

Offset: 0

Paul Curtz, Jul 16 2008

Also: The numerators in the j=2 column of the array a(i,j) defined in A140825, where the columns j=0 and j=1 are represented by A000012 and A005408. This could be extended to column j=3: 1, -1, 9, 55, 161, ... The common feature of these sequences derived from a(i,j) is that their j-th differences are constant sequences defined by A091137(j).
a(n) is the set of all k such that 6*k + 6 is a perfect square. - Gary Detlefs, Mar 04 2010
The identity (6*n^2 - 1)^2 - (9*n^2 - 3)*(2*n)^2 = 1 can be written as a(n+1)^2 - A157872(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012
Apart from first term, sequence found by reading the line from 5, in the direction 5, 23, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
From Paul Curtz, Sep 17 2018: (Start)
Terms from center to right in the following spiral:
.
                    65--63--61--59
                    /             \
                  67  31--29--27  57
                  /   /         \   \
                69  33   9---7  25  55
                /   /   /     \   \   \
              71  35  11  -1===5==23==53==>
                  /   /   /   /   /   /
                37  13   1---3  21  51
                  \   \         /   /
                  39  15--17--19  49
                    \             /
                    41--43--45--47 (End)

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969, 132 pages, pp. 28-36. CCSA, then CELAR. Now DGA Maitrise de l'Information 35131 Bruz.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Barred Stars.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A157872, A060747, A103115(n+1), A141417 (array).
Cf. A000012, A001318, A005408, A091137, A140825.
Cf. A003154, A032528, A033581, A016777, A017593.

[6*n^2 - 1: n in [0..50]]; // Vincenzo Librandi, Jun 02 2011

LinearRecurrence[{3,-3,1},{-1,5,23},40] (* Vincenzo Librandi, Feb 05 2012 *)
CoefficientList[Series[(1-8*x-5*x^2)/(x-1)^3 , {x, 0, 40}], x] (* Stefano Spezia, Sep 17 2018 *)

a(n)=6*n^2-1 \\ Charles R Greathouse IV, Jun 01 2011

a(n) = 2*a(n-1) - a(n-2) + 12.
First differences: a(n+1) - a(n) = A017593(n).
Second differences: A071593(n+1) - A071593(n) = 12.
G.f.: (1-8*x-5*x^2)/(x-1)^3. - Jaume Oliver Lafont, Aug 30 2009
From Vincenzo Librandi, Feb 05 2012: (Start)
a(n) = a(n-1) + 12*n - 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A033581(n) - 1. - Omar E. Pol, Jul 18 2012
a(n) = A032528(2*n) - 1. - Adriano Caroli, Jul 21 2013
For n > 0, a(n) = floor(3/(cosh(1/n) - 1)) = floor(1/(n*sinh(1/n) - 1)); for similar formulas for cosine and sine, see A033581. - Clark Kimberling, Oct 19 2014, corrected by M. F. Hasler, Oct 21 2014
a(-n) = a(n). - Paul Curtz, Sep 17 2018
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(6))*cot(Pi/sqrt(6)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(6))*csc(Pi/sqrt(6)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(6))*csc(Pi/sqrt(6)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(6))*sin(Pi/sqrt(3))/sqrt(2). (End)
a(n) = A003154(n+1) - 2*A016777(n). - Leo Tavares, May 13 2022
E.g.f.: exp(x)*(6*x^2 + 6*x - 1). - Elmo R. Oliveira, Jan 16 2025

Edited and extended by R. J. Mathar, Aug 06 2008

Better description Ray Chandler, Feb 03 2009

A195189 Denominators of a sequence leading to gamma = A001620.

Original entry on oeis.org

2, 24, 72, 2880, 800, 362880, 169344, 29030400, 9331200, 4790016000, 8673280, 31384184832000, 6181733376000, 439378587648000, 10346434560000, 512189896458240000, 265423814656, 14148260909088768000, 2076423318208512000, 96342919523794944000000, 74538995631567667200000

Offset: 0

Paul Curtz, Sep 11 2011

gamma = 1/2 + 1/24 + 1/72 + 19/2880 + 3/800 + 863/362880 + 275/169344 + ... = (A002206 unsigned=reduced A141417(n+1)/A091137(n+1))/a(n) is an old formula based on Gregory's A002206/A002207.

This formula for Euler's constant was discovered circa 1780-1790 by the Italian mathematicians Gregorio Fontana (1735-1803) and Lorenzo Mascheroni (1750-1800), and was subsequently rediscovered several times (in particular, by Ernst Schröder in 1879, Niels E. Nørlund in 1923, Jan C. Kluyver in 1924, Charles Jordan in 1929, Kenter in 1999, and Victor Kowalenko in 2008). For more details, see references below. - Iaroslav V. Blagouchine, May 03 2015

			a(0)=1*2, a(1)=2*12, a(2)=3*24, a(3)=4*720.

G. C. Greubel, Table of n, a(n) for n = 0..440
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version.
M. Coffey and J. Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, Acta Appl. Math., 121 (2012), 1-3.
J. C. Kluyver, Euler's constant and natural numbers, Proc. Kon. Ned. Akad. Wet., 27(1-2) (1924), 142-144.

Cf. A001620, A002206, A002207, A091137, A141417.

g[n_]:=Sum[StirlingS1[n,l]/(l+1),{l,1,n}]/(n*n!); a[n_]:=Denominator[g[n]]; Table[a[n],{n,1,30}] (* Iaroslav V. Blagouchine, May 03 2015 *)
g[n_] := Sum[ BernoulliB[j]/j * StirlingS1[n, j-1], {j, 1, n+1}] / n! ; a[n_] := (n+1)*Denominator[g[n]]; Table[a[n], {n, 0, 20}]
(* or *) max = 20; Denominator[ CoefficientList[ Series[ 1/Log[1 + x] - 1/x, {x, 0, max}], x]]*Range[max+1] (* Jean-François Alcover, Sep 04 2013 *)

a(n) = (n+1) * A002207(n).

More terms from Jean-François Alcover, Sep 04 2013

A007525 Decimal expansion of log_2 e.

Original entry on oeis.org

1, 4, 4, 2, 6, 9, 5, 0, 4, 0, 8, 8, 8, 9, 6, 3, 4, 0, 7, 3, 5, 9, 9, 2, 4, 6, 8, 1, 0, 0, 1, 8, 9, 2, 1, 3, 7, 4, 2, 6, 6, 4, 5, 9, 5, 4, 1, 5, 2, 9, 8, 5, 9, 3, 4, 1, 3, 5, 4, 4, 9, 4, 0, 6, 9, 3, 1, 1, 0, 9, 2, 1, 9, 1, 8, 1, 1, 8, 5, 0, 7, 9, 8, 8, 5, 5, 2, 6, 6, 2, 2, 8, 9, 3, 5, 0, 6, 3, 4, 4, 4, 9, 6, 9, 9

Offset: 1

N. J. A. Sloane

Around 1670, James Gregory discovered by inversion of 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... = log(2) that 1 + 1/2 - 1/12 + 1/24 - 19/720 + (27/1440 = 3/160) - 863/60480 + ... = 1/log(2). See formula with A002206 and A002207. See also A141417 signed /A091137; case i = 0 in A165313. First row in array p. 36 of the reference. - Paul Curtz, Sep 12 2011

This constant 1/log(2) is also related to the asymptotic evaluation of the maximum number of subtraction steps required to compute gcd(m, n) by the binary Euclidean algorithm, m and n being odd and chosen at random. - Jean-François Alcover, Jun 23 2014, after Steven Finch

			1.442695040888963407359924681...

Paul Curtz, Intégration numérique des systèmes différentiels .. , note n° 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.18 Porter-Hensley constants, p. 159.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 25, equation 25:14:3 at page 232.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Simon Plouffe, 1/log(2) the inverse of the natural logarithm of 2.
Srinivasa Ramanujan, Question 769, J. Ind. Math. Soc.
Index entries for transcendental numbers.

Cf. A000670, A002206, A002207, A052882, A091137, A141417, A165313.

RealDigits[N[1/Log[2], 105]][[1]] (* Jean-François Alcover, Oct 30 2012 *)

1/log(2) \\ Charles R Greathouse IV, Jan 04 2016

Equals lim_{n->infinity} A000670(n)/A052882(n). - Mats Granvik, Aug 10 2009
Equals Sum_{k>=-1} A002206(k)/A002207(k). - Paul Curtz, Sep 12 2011
Also equals integral_{x>=2} 1/(x*log(x)^2). - Jean-François Alcover, May 24 2013
1/log(2) = Sum_{n = -infinity..infinity} (2^n / (1 + 2^2^n)). - Nicolas Nagel, Mar 16 2018
More generally: 1/log(2) = Sum_{n = -infinity..infinity} (2^(n+x) / (1 + 2^2^(n+x))) for all real x. - Nicolas Nagel, Jul 02 2019
From Amiram Eldar, Jun 04 2023: (Start)
Equals 1 + Sum_{k>=1} 1/(2^k * (1 + 2^(1/2^k))).
Equals Product_{k>=1} ((1 + 2^(1/2^k))/2). (End)

A141412 Triangle c(n,k) of the denominators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 12, 2, 1, 5, 6, 4, 1, 1, 6, 180, 8, 6, 2, 1, 7, 10, 15, 2, 6, 1, 1, 8, 560, 240, 240, 6, 4, 2, 1, 9, 1260, 15120, 20, 144, 1, 12, 1, 1, 10, 12600, 672, 945, 32, 240, 8, 3, 2, 1, 11, 1260, 8400, 1512, 3024, 48, 240, 3, 1, 1, 1, 12, 166320, 100800, 64800, 12096, 12096, 480, 360, 4, 12, 2, 1

Offset: 0

Paul Curtz, Aug 04 2008

Polynomials are characteristic polynomials of a particular John Couch Adams matrix.
General term: ( (-1)^(n-j)*C(j, n)*n! ) * Integral_{0..i} (u*(u-1)*(u-2)* ... *(u-n))/(u-j)) du, with 1 <= i,j <= n (see Flajolet et al.).
Denominators are 1, 2, 12, 24, 720 = A091137.
These polynomials come from the explicit case. The less interesting implicit case has the same denominators (see P. Curtz reference).

			Triangle begins:
  1;
  2,   1;
  3,   1,  1;
  4,  12,  2,  1;
  5,   6,  4,  1,  1;
  6, 180,  8,  6,  2,  1;
  7,  10, 15,  2,  6,  1,  1;
  ...

Paul Curtz, Intégration .. note 12, C.C.S.A., Arcueil 1969, p. 61; ibid. pp. 62-65.
P. Flajolet, X. Gourdon, and B. Salvy, Sur une famille de polynômes issus de l'analyse numérique, Gazette des Mathématiciens, 1993, 55, pp. 67-78.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Bakir Farhi, On the derivatives of the integer-valued polynomials, arXiv:1810.07560 [math.NT], 2018.

Cf. A000254, A048594, A129891, A140749 (numerators).

[Denominator(Factorial(k)*StirlingFirst(n, k)/Factorial(n)): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 24 2023

P := proc(n,x) option remember ; if n =0 then 1; else (-1)^n/(n+1)+x*add( (-1)^i/(i+1)*procname(n-1-i,x),i=0..n-1) ; expand(%) ; fi; end:
A141412 := proc(n,k) p := P(n,x) ; denom(coeftayl(p,x=0,k)) ; end: seq(seq(A141412(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Aug 24 2009

p[0]=1; p[n_]:= p[n]= (-1)^n/(n+1) +x*Sum[(-1)^k*p[n-1-k]/(k+1), {k, 0, n-1}];
Denominator[Flatten[Table[CoefficientList[p[n], x], {n,0,11}]]][[1 ;; 72]] (* Jean-François Alcover, Jun 17 2011 *)
Table[Denominator[(k+1)!*StirlingS1[n+1,k+1]/(n+1)!], {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, Oct 24 2023 *)

def A141412(n,k): return denominator(factorial(k+1)* stirling_number1(n+1,k+1)/factorial(n+1))
flatten([[A141412(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

Conjecture: T(n, k) = d(n+1, k+1), with d(n,k) = denominator(A000254(n, k)*k!/n!) where A000254 are the unsigned Stirling numbers of the 1st kind. See d(n,k) in Farhi link. - Michel Marcus, Oct 18 2018

Equals denominators of A048594(n+1, k+1)/(n+1)!. - G. C. Greubel, Oct 24 2023

Partially edited by R. J. Mathar, Aug 24 2009

A171080 a(n) = Product_{3 <= p <= 2n+1, p prime} p^floor(2n / (p - 1)).

Original entry on oeis.org

1, 3, 45, 945, 14175, 467775, 638512875, 1915538625, 488462349375, 194896477400625, 32157918771103125, 2218896395206115625, 3028793579456347828125, 9086380738369043484375, 3952575621190533915703125, 28304394023345413370350078125, 7217620475953080409439269921875

Offset: 0

N. J. A. Sloane, Sep 06 2010

nonn

F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 3rd. ed., 1966; Lemma 1.5.2, p. 13.

Victor M. Buchstaber, Alexander P. Veselov, Todd polynomials and Hirzebruch numbers, arXiv:2310.07383 [math.AT], Oct. 2023.

Cf. A091137, A238963, A160014.

f:=proc(n) local q,t1; t1:=1; for q from 3 to 2*n+1 do if isprime(q) then t1:=t1*q^floor(2*n/(q-1)); fi; od; t1; end;

a[n_] := Product[If[PrimeQ[q], q^Floor[2 n/(q - 1)], 1], {q, 3, 2 n + 1}]
Table[a[n], {n, 0, 20}] (* Wolfgang Hintze, Oct 03 2014 *)

from functools import cache
@cache
def a_rec(n):
    if n == 0: return 1
    p = mul(s for s in map(lambda i: i+1, divisors(2*n)) if is_prime(s))
    return (p * a_rec(n - 1)) // 2
print([a_rec(n) for n in range(17)])  # Peter Luschny, Dec 12 2023

From Peter Luschny, Dec 12 2023: (Start)
a(n) = (Clausen(2*n)*a(n-1))/2 for n > 0, where Clausen(n) = A160014(1, n).
a(n) = A091137(2*n) / 2^(2*n).  (End)

A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

Original entry on oeis.org

1, 1, 7, 17, 41, 731, 8563, 27719, 190073, 516149, 1013143139, 1519024289, 14108351869, 14399405173, 23142912688967, 83945247395407, 84894728616107, 3204549982389941, 262488267575333123, 9027726081126601799, 2026692221793223022131, 1375035304877251309001

Offset: 0

Paul Curtz, Aug 27 2011

Akiyama-Tanigawa from 1/n gives Bernoulli A164555(n)/A027642(n).
Reciprocally
1,   1/2,   5/12,     3/8, 251/720,    95/288, 19087/60480, 5257/17280,
1/2, 1/6,    1/8,  19/180,    3/32, 863/10080,    275/3456,
1/3, 1/12, 7/120,  17/360, 41/1008, 731/20160, 8563/259200,
1/4, 1/20, 1/30,   11/420, 89/4032,5849/302400,
1/5, 1/30, 3/140, 83/5040, 59/4320,
1/6, 1/42, 5/336,
1/7, 1/56,
1/8.
First row: A002208/A002209 or reduced A002657(n)/A091137(n) unsigned.
Second row: A002206(n+1)/A002689(n) unsigned. See A141417(n) and A174727(n).
Third row: a(n)/A194506(n).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.

Cf. A194506 (denominator).

read("transforms3") ;
L := [seq(1/n,n=1..20)] ;
L1 := AKIYAMATANIGAWAi(L) ;
L2 := AKIYATANI(L1) ;
L3 := AKIYATANI(L2) ;
apply(numer,%) ; # R. J. Mathar, Aug 27 2011
# second Maple program:
b:= proc (n, k) option remember;
      `if`(n=0, 1/(k+1), b(n-1, k) -b(n-1, k+1)/n)
    end:
a:= n-> numer(b(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 27 2011

a[n_, 0] := 1/(n+1); a[n_, m_] := a[n, m] = a[n, m-1] - a[n+1, m-1]/m; Table[a[2, m], {m, 0, 21}] // Numerator (* Jean-François Alcover, Aug 09 2012 *)
Numerator@Table[(-1)^n (n + 1) Integrate[FunctionExpand[x Binomial[x, n + 1]], {x, 0, 1}], {n, 0, 20}] (* Vladimir Reshetnikov, Feb 01 2017 *)

a(n)/A194506(n) = (-1)^n * (n+1) * Integral_{0Vladimir Reshetnikov, Feb 01 2017

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