A285864 Triangle read by rows: a(n,m) = numerator(binomial(n,m)*2^(n-m)*B(n-m)) with B(k) the Bernoulli numbers A027641(k)/A027642(k).

1, -1, 1, 2, -2, 1, 0, 2, -3, 1, -8, 0, 4, -4, 1, 0, -8, 0, 20, -5, 1, 32, 0, -8, 0, 10, -6, 1, 0, 32, 0, -56, 0, 14, -7, 1, -128, 0, 128, 0, -112, 0, 56, -8, 1, 0, -384, 0, 128, 0, -336, 0, 24, -9, 1, 2560, 0, -384, 0, 320, 0, -112, 0, 30, -10, 1

Offset: 0

Author

Wolfdieter Lang, May 03 2017

Keywords

• sign
• easy
• tabl
• frac

Comments

The denominator triangle b(n,m) is given in A285865.

a(n,m)/b(n,m) = B(2;n,m) is the d = 2 instance of the fractional d-family of triangles B(d;n,m) = binomial(n,m)*d^(n-m)*B(n-m), for d >= 1. They are the coefficient triangles of generalized Bernoulli polynomials PB(d;n,x) = Sum_{m=0..n} B(d;n,m)*x^m for n >= 0.

{PB(d;n,x)}{n>=0} has e.g.f. EB(d;x,z) := Sum{n>=0} PB(d;n,x)*z^n = d*z*exp(x*z)/(exp(d*z)-1). B(d;n,m) is a Sheffer triangle of the Appell type for each d, denoted by (d*z/(exp(d*z - 1)), z).

PB(d;n,x) gives a (trivial) generalization of the Bernoulli polynomials with coefficients given in A196838/A196839 (rising powers of x), and this is PB(1;n,x).

The polynomials PB(d;n,x) appear in the generalized Faulhaber formula for sums of powers of arithmetic progressions SP(n,m) := Sum_{j=0..m} (a + d*j)^n, n >= 0, m >= 0, d >= 1, a = 0 for d = 1 and a from the smallest positive restricted residue system modulo d >= 2. For this Faulhaber formula see a comment in A285863, where they are named B(d;n,x).

The row sums of the rational triangle B(2;n,m) give A157779(n)/A141459(n). The alternating row sums are given in A285866/A141459(n).

Examples

			The triangle a(n,m) begins:
n\m    0    1    2   3    4    5    6  7  8   9 10 ...
0:     1
1:    -1    1
2:     2   -2    1
3:     0    2   -3   1
4:    -8    0    4  -4    1
5:     0   -8    0  20   -5    1
6:    32    0   -8   0   10   -6    1
7:     0   32    0 -56    0   14   -7  1
8:  -128    0  128   0 -112    0   56 -8  1
9:     0 -384    0 128    0 -336    0 24 -9   1
10: 2560    0 -384   0  320    0 -112  0 30 -10  1
...
The rational triangle B(2;n,m) = a(n,m)/A285865(n,m) begins:
n\m     0       1        2     3     4      5     6    7    8   9  10 ...
0:      1
1:     -1       1
2:     2/3     -2        1
3:      0       2       -3     1
4:    -8/15     0        4    -4     1
5:      0     -8/3       0   20/3   -5      1
6:    32/21     0       -8     0    10     -6     1
7:      0     32/3       0  -56/3    0     14    -7    1
8:  -128/15     0      128/3   0  -112/3    0   56/3  -8    1
9:      0    -384/5      0    128    0   -336/5   0   24   -9   1
10:  2560/33    0      -384    0    320     0   -112   0   30 -10   1
...
		

Crossrefs

Cf. A027641/A027642, A157779/A141459, A196838/A196839, A285863, A285865.

Programs

Maple

T := d -> (n,m) -> numer(binomial(n, m)*d^(n-m)*bernoulli(n-m)):
for n from 0 to 10 do seq(T(2)(n,k),k=0..n) od; # Peter Luschny, May 04 2017

Mathematica

T[n_, m_]:=Numerator[Binomial[n, m]*2^(n - m)*BernoulliB[n - m]]; Table[T[n, m], {n, 0, 20}, {m, 0, n}] // Flatten (* Indranil Ghosh, May 06 2017 *)

PARI

T(n, m) = numerator(binomial(n, m)*2^(n - m)*bernfrac(n - m));
for(n=0, 20, for(m=0, n, print1(T(n, m),", ");); print();) \\ Indranil Ghosh, May 06 2017

Python

from sympy import binomial, bernoulli
def T(n, m): return (binomial(n, m) * (-2)**(n - m) * bernoulli(n - m)).numerator
for n in range(21): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, May 06 2017

Formula

a(n,m) = numerator(binomial(n, m)*2^(n-m)*B(n-m)), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).

E.g.f.s of the rational column sequences {B(2;n, m)}_{n>=0} are Ecol(m, x) = (2*x/(exp(2*x) - 1))*x^m/m! (Sheffer property). Here the numerators of column m are numerator([x^m/m!] Ecol(m, x)), m >= 0.