A290317 -id:A290317

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A051716

Numerators of Bernoulli twin numbers C(n).

+10
22

1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, -7, -3617, 3617, 43867, -43867, -174611, 174611, 854513, -854513, -236364091, 236364091, 8553103, -8553103, -23749461029, 23749461029, 8615841276005, -8615841276005, -7709321041217, 7709321041217, 2577687858367

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,11

COMMENTS

The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to Paul Curtz.

For denominators see A051717.

Negatives of numerators of column 1 of table described in A051714/A051715.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..640

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

FORMULA

Numerators of differences of the sequence of rational numbers 0 followed by A164555/A027642. - Paul Curtz, Jan 29 2017

The e.g.f. of the rationals a(n)/A051717(n) is -(1/x + x^2/2 + x/(1 - exp(x)) + dilog(exp(-x))), (with dilog(x) = polylog(2, 1-x)). From integrating the e.g.f. of the z-sequence (exp(x) - (1+x))/(exp(x) -1)^2 for the Bernoulli polynomials of the second kind (A290317 / A290318). - Wolfdieter Lang, Aug 07 2017

EXAMPLE

The C(n) sequence is 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...

MAPLE

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;

MATHEMATICA

c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Numerator[c[n]], {n, 0, 34}] (* Jean-François Alcover, Dec 19 2011 *)

PROG

(PARI) a(n) = if (n==0, 1, nu = numerator(bernfrac(n)+bernfrac(n-1)); if (n%2, -nu, nu)); \\ Michel Marcus, Jan 29 2017

(Magma)

f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;

function A051716(n)

if n eq 0 then return 1;

elif (n mod 2) eq 0 then return Numerator(f(n));

else return Numerator(-f(n));

end if;

end function;

[A051716(n): n in [0..50]]; // G. C. Greubel, Apr 22 2023

(SageMath)

def f(n): return bernoulli(n)+bernoulli(n-1)

def A051716(n):

if (n==0): return 1

elif (n%2==0): return numerator(f(n))

else: return numerator(-f(n))

[A051716(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

CROSSREFS

Cf. A000367, A027641, A027642, A051714, A051715.

Cf. A051717, A129825, A129826, A129724, A164555.

KEYWORD

sign,easy,nice,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Dec 08 1999

Edited by N. J. A. Sloane, May 25 2008

STATUS

approved

A286718

Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.

+10
13

1, 1, 1, 4, 5, 1, 28, 39, 12, 1, 280, 418, 159, 22, 1, 3640, 5714, 2485, 445, 35, 1, 58240, 95064, 45474, 9605, 1005, 51, 1, 1106560, 1864456, 959070, 227969, 28700, 1974, 70, 1, 24344320, 42124592, 22963996, 5974388, 859369, 72128, 3514, 92, 1, 608608000, 1077459120, 616224492, 172323696, 27458613, 2662569, 159978, 5814, 117, 1, 17041024000, 30777463360, 18331744896, 5441287980, 941164860, 102010545, 7141953, 322770, 9090, 145, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

This is a generalization of the unsigned Stirling1 triangle A132393.

In general the lower triangular Sheffer matrix ((1 - d*x)^(-a/d), (-1/d)*log(1 - d*x)) is called here |S1hat[d,a]|. The signed matrix S1hat[d,a] with elements (-1)^(n-k)*|S1hat[d,a]|(n, k) is the inverse of the generalized Stirling2 Sheffer matrix S2hat[d,a] with elements S2[d,a](n, k)/d^k, where S2[d,a] is Sheffer (exp(a*x), exp(d*x) - 1).

In the Bala link the signed S1hat[d,a] (with row scaled elements S1[d,a](n,k)/d^n where S1[d,a] is the inverse matrix of S2[d,a]) is denoted by s_{(d,0,a)}, and there the notion exponential Riordan array is used for Sheffer array.

In the Luschny link the elements of |S1hat[m,m-1]| are called Stirling-Frobenius cycle numbers SF-C with parameter m.

From Wolfdieter Lang, Aug 09 2017: (Start)

The general row polynomials R(d,a;n,x) = Sum_{k=0..n} T(d,a;n,k)*x^k of the Sheffer triangle |S1hat[d,a]| satisfy, as special polynomials of the Boas-Buck class (see the reference), the identity (we use the notation of Rainville, Theorem 50, p. 141, adapted to an exponential generating function)

(E_x - n*1)*R(d,a;n,x) = -n!*Sum_{k=0..n-1} d^k*(a*1 + d*beta(k)*E_x)*R(d,a;n-1-k,x)/(n-1-k)!, for n >= 0, with E_x = x*d/dx (Euler operator), and beta(k) = A002208(k+1)/A002209(k+1).

This entails a recurrence for the sequence of column k, for n > k >= 0: T(d,a;n,k) = (n!/(n - k))*Sum_{p=k..n-1} d^(n-1-p)*(a + d*k*beta(n-1-p))*T(d,a;p,k)/p!, with input T(d,a;k,k) = 1. For the present [d,a] = [3,1] case see the formula and example sections below. (End)

The inverse of the Sheffer triangular matrix S2[3,1] = A282629 is the Sheffer matrix S1[3,1] = (1/(1 + x)^(1/3), log(1 + x)/3) with rational elements S1[3,1](n, k) = (-1)^(n-m)*T(n, k)/3^n. - Wolfdieter Lang, Nov 15 2018

REFERENCES

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).

Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.

LINKS

Table of n, a(n) for n=0..65.

P. Bala, A 3 parameter family of generalized Stirling numbers

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.

Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Peter Luschny, The Stirling-Frobenius numbers.

FORMULA

Recurrence: T(n, k) = T(n-1, k-1) + (3*n-2)*T(n-1, k), for n >= 1, k = 0..n, and T(n, -1) = 0, T(0, 0) = 1 and T(n, k) = 0 for n < k.

E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k (i.e., e.g.f. of the triangle) is (1 - 3*z)^{-(x+1)/3}.

E.g.f. of column k is (1 - 3*x)^(-1/3)*((-1/3)*log(1 - 3*x))^k/k!.

Recurrence for row polynomials is R(n, x) = (x+1)*R(n-1, x+3), with R(0, x) = 1.

Row polynomial R(n, x) = risefac(3,1;x,n) with the rising factorial

risefac(d,a;x,n) := Product_{j=0..n-1} (x + (a + j*d)). (For the signed case see the Bala link, eq. (16)).

T(n, k) = sigma^{(n)}_{n-k}(a_0,a_1,...,a_{n-1}) with the elementary symmetric functions with indeterminates a_j = 1 + 3*j.

T(n, k) = Sum_{j=0..n-k} binomial(n-j, k)*|S1|(n, n-j)*3^j, with the unsigned Stirling1 triangle |S1| = A132393.

Boas-Buck column recurrence (see a comment above): T(n, k) =

(n!/(n - k))*Sum_{p=k..n-1} 3^(n-1-p)*(1 + 3*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, with beta(k) = A002208(k+1)/A002209(k+1). See an example below. - Wolfdieter Lang, Aug 09 2017

EXAMPLE

The triangle T(n, k) begins:

n\k 0 1 2 3 4 5 6 7 8 ...

O: 1

1: 1 1

2: 4 5 1

3: 28 39 12 1

4: 280 418 159 22 1

5: 3640 5714 2485 445 35 1

6: 58240 95064 45474 9605 1005 51 1

7: 1106560 1864456 959070 227969 28700 1974 70 1

8: 24344320 42124592 22963996 5974388 859369 72128 3514 92 1

...

From Wolfdieter Lang, Aug 09 2017: (Start)

Recurrence: T(3, 1) = T(2, 0) + (3*3-2)*T(2, 1) = 4 + 7*5 = 39.

Boas-Buck recurrence for column k = 2 and n = 5:

T(5, 2) = (5!/3)*(3^2*(1 + 6*(3/8))*T(2,2)/2! + 3*(1 + 6*(5/12)*T(3, 2)/3! + (1 + 6*(1/2))* T(4, 2)/4!)) = (5!/3)*(9*(1 + 9/4)/2 + 3*(1 + 15/6)*12/6 + (1 + 3)*159/24) = 2485.

The beta sequence begins: {1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, ...}.

(End)

MATHEMATICA

T[n_ /; n >= 1, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (3*n-2)* T[n-1, k]; T[_, -1] = 0; T[0, 0] = 1; T[n_, k_] /; n<k = 0;

Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018 *)

CROSSREFS

S2[d,a] for [d,a] = [1,0], [2,1], [3,1], [3,2], [4,1] and [4,3] is A048993, A154537, A282629, A225466, A285061 and A225467, respectively.

S2hat[d,a] for these [d,a] values is A048993, A039755, A111577 (offset 0), A225468, A111578 (offset 0) and A225469, respectively.

|S1hat[d,a]| for [d,a] = [1,0], [2,1], [3,2], [4,1] and [4,3] is A132393, A028338, A225470, A290317 and A225471, respectively.

Column sequences for k = 0, 1: A007559, A024216.

Diagonal sequences: A000012, A000326(n+1), A024212(n+1), A024213(n+1).

Row sums: A008544. Alternating row sums: A000007.

Beta sequence: A002208(n+1)/A002209(n+1).

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, May 18 2017

STATUS

approved

A290318

Triangle read by rows. Row n gives the denominators of the coefficients of the Bernoulli polynomials of the second kind (in rising powers).

+10
2

1, 2, 1, 6, 1, 1, 4, 1, 2, 1, 30, 1, 1, 1, 1, 4, 1, 1, 3, 2, 1, 84, 1, 1, 1, 2, 1, 1, 24, 1, 1, 3, 4, 1, 2, 1, 90, 1, 1, 1, 1, 1, 3, 1, 1, 20, 1, 1, 1, 1, 5, 1, 1, 2, 1, 132, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The numerators appear in A290317, where more information is found.

The LCM of row n seems to be A002790(n).

The first column is given by A006233.

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

The triangle T(n, k) begins:

n\k 0 1 2 3 4 5 6 7 8 9 10 ...

0: 1

1: 2 1

2: 6 1 1

3: 4 1 2 1

4: 30 1 1 1 1

5: 4 1 1 3 2 1

6: 84 1 1 1 2 1 1

7: 24 1 1 3 4 1 2 1

8: 90 1 1 1 1 1 3 1 1

9: 20 1 1 1 1 5 1 1 2 1

10: 132 1 1 1 1 1 1 1 2 1 1

...

For the first rows of the rational triangle r(n, m) see A290317.

CROSSREFS

Cf. A002790, A006233, A290317 (numerators).

KEYWORD

nonn,easy,frac,tabl

AUTHOR

Wolfdieter Lang, Aug 06 2017

EXTENSIONS

Definition corrected by Eric M. Schmidt, Dec 16 2017

STATUS

approved

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