OFFSET
0,4
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).
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.
EXAMPLE
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
...
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, May 03 2017
STATUS
approved