A285865 -id:A285865 - cp's OEIS frontend

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

Wolfdieter Lang, May 03 2017

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).

			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
...

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

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

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 *)

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

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

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.

A285866 a(n) = numerator((-2)^nSum_{k=0..n} binomial(n,k) Bernoulli(k, 1/2)).

Original entry on oeis.org

1, -2, 11, -6, 127, -10, 221, -14, 367, -18, -1895, -22, 1447237, -26, -57253, -30, 118526399, -34, -5749677193, -38, 91546283957, -42, -1792042789427, -46, 1982765468376757, -50, -286994504449237, -54, 3187598676787485443, -58, -4625594554880206360895, -62

Offset: 0

Wolfdieter Lang, May 03 2017

Previous name: Numerators of alternating row sums of the rational triangle B2 = A285864/A285865.

The denominators are given in A141459.

Cf. A027641/A027642, A141459, A285864/A285865.

See also A157781/A157782 and A157811/A285068.

a := n -> numer((-2)^n*add(binomial(n,k)*bernoulli(k,1/2), k=0..n)):
seq(a(n), n=0..31); # Peter Luschny, Jul 24 2020

a[n_] := (-2)^n Sum[Binomial[n, k] BernoulliB[k, 1/2], {k, 0, n}] // Numerator;
Table[a[n], {n, 0, 31}] (* Peter Luschny, Jul 24 2020 *)

# uses [gen_bernoulli_number from A157811]
print([numerator((-1)^n*gen_bernoulli_number(n, 2)) for n in range(33)]) # Peter Luschny, Mar 26 2021

a(n) = numerator(Sum_{m=0..n} (-1)^m*A285864(n, m)/A285865(n, m)), n >= 0, where the rational triangle is B2(n, m) = binomial(m, m)*2^(n-m)*B(n-m), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).

More terms from Indranil Ghosh, May 06 2017

New name by Peter Luschny, Jul 24 2020

A336454 a(n) = denominator(2^nSum_{k=0..n} binomial(n, k)Bernoulli(k, 1/2)*x^(n-k)).

Original entry on oeis.org

1, 1, 3, 1, 15, 3, 21, 3, 15, 5, 33, 3, 1365, 105, 15, 3, 255, 15, 1995, 105, 1155, 165, 345, 15, 1365, 273, 21, 7, 435, 15, 7161, 231, 19635, 1785, 105, 3, 959595, 25935, 1365, 105, 47355, 1155, 49665, 1155, 2415, 2415, 4935, 105, 23205, 3315, 7293, 429, 8745

Offset: 0

Peter Luschny, Jul 24 2020

Cf. A336517/A285865 (coefficients of polynomials).

Bcp := n -> 2^n*add(binomial(n, k)*bernoulli(k, 1/2)*x^(n-k), k=0..n):
a := n -> denom(Bcp(n)): seq(a(n), n=0..52);

A336517 T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^nSum_{k=0..n} binomial(n, k) Bernoulli(k, 1/2) * x^(n-k). Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, -1, 0, 4, 0, -2, 0, 8, 7, 0, -8, 0, 16, 0, 14, 0, -80, 0, 32, -31, 0, 28, 0, -80, 0, 64, 0, -62, 0, 392, 0, -224, 0, 128, 127, 0, -496, 0, 1568, 0, -1792, 0, 256, 0, 762, 0, -992, 0, 9408, 0, -1536, 0, 512, -2555, 0, 1524, 0, -4960, 0, 6272, 0, -3840, 0, 1024

Offset: 0

Peter Luschny, Jul 24 2020

Consider polynomials B_a(n, x) = a^n*Sum_{k=0..n} binomial(n, k)*Bernoulli(k, 1/a)*x^(n - k), with a != 0. They form an Appell sequence, the case a = 1 are the Bernoulli polynomials. T(n, k) are the numerators of the coefficients of the polynomials in the case a = 2.

			Rational polynomials start, coefficients of [numerators | denominators]:
                                           [ [1], [ 1]]
                                       [[0,   2], [ 1, 1]]
                                   [[-1, 0,   4], [ 3, 1, 1]]
                             [[0,    -2, 0,   8], [ 1, 1, 1, 1]]
                          [[7, 0,    -8, 0,  16], [15, 1, 1, 1, 1]]
                    [[0,   14, 0,   -80, 0,  32], [ 1, 3, 1, 3, 1, 1]]
               [[-31, 0,   28, 0,   -80, 0,  64], [21, 1, 1, 1, 1, 1, 1]]
           [[0,  -62, 0,  392, 0,  -224, 0, 128], [ 1, 3, 1, 3, 1, 1, 1, 1]]
      [[127, 0, -496, 0, 1568, 0, -1792, 0, 256], [15, 1, 3, 1, 3, 1, 3, 1, 1]]
   [[0, 762, 0, -992, 0, 9408, 0, -1536, 0, 512], [ 1, 5, 1, 1, 1, 5, 1, 1, 1, 1]]

Cf. A285865 (denominators), A336454 (polynomial denominator), A141459, A157779, A285866.

Bcp := n -> 2^n*add(binomial(n,k)*bernoulli(k,1/2)*x^(n-k), k=0..n):
polycoeff := p -> seq(numer(coeff(p, x, k)), k = 0..degree(p, x)):
Trow := n -> polycoeff(Bcp(n)): seq(Trow(n), n=0..10);

Denominator(b(n, 1)) = A141459(n).
Numerator(b(n, -1)) = A285866(n).
Numerator(b(n, 0)) = A157779(n).

A335953 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)2^k Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 7, 0, -2, 0, 1, 0, 7, 0, -10, 0, 1, -31, 0, 7, 0, -5, 0, 1, 0, -31, 0, 49, 0, -7, 0, 1, 127, 0, -124, 0, 98, 0, -28, 0, 1, 0, 381, 0, -124, 0, 294, 0, -12, 0, 1, -2555, 0, 381, 0, -310, 0, 98, 0, -15, 0, 1

Offset: 0

Peter Luschny, Jul 25 2020

			[0]   1
[1]   0,   1
[2]  -1,   0,    1
[3]   0,  -1,    0,    1
[4]   7,   0,   -2,    0,  1
[5]   0,   7,    0,  -10,  0,   1
[6] -31,   0,    7,    0, -5,   0,   1
[7]   0, -31,    0,   49,  0,  -7,   0,   1
[8] 127,   0, -124,    0, 98,   0, -28,   0, 1
[9]   0, 381,    0, -124,  0, 294,   0, -12, 0, 1

Peter H. N. Luschny, An introduction to the Bernoulli function, arXiv:2009.06743 [math.HO], 2020.

Cf. A285865 (denominators), A336454 (polynomial denominator), A336517, A001896, A001897.

Bcn := n -> 2^n*bernoulli(n, 1/2):
Bcp := n -> add(binomial(n, k)*Bcn(k)*x^(n-k), k=0..n):
polycoeff := p -> seq(numer(coeff(p, x, k)), k = 0..degree(p, x)):
Trow := n -> polycoeff(Bcp(n)): seq(print(Trow(n)), n=0..9);

cp's OEIS Frontend

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).

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A285866 a(n) = numerator((-2)^n*Sum_{k=0..n} binomial(n,k) * Bernoulli(k, 1/2)).

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A336454 a(n) = denominator(2^n*Sum_{k=0..n} binomial(n, k)*Bernoulli(k, 1/2)*x^(n-k)).

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A336517 T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^n*Sum_{k=0..n} binomial(n, k) * Bernoulli(k, 1/2) * x^(n-k). Triangle read by rows, for 0 <= k <= n.

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A335953 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)*2^k* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

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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).

A285866 a(n) = numerator((-2)^nSum_{k=0..n} binomial(n,k) Bernoulli(k, 1/2)).

A336454 a(n) = denominator(2^nSum_{k=0..n} binomial(n, k)Bernoulli(k, 1/2)*x^(n-k)).

A336517 T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^nSum_{k=0..n} binomial(n, k) Bernoulli(k, 1/2) * x^(n-k). Triangle read by rows, for 0 <= k <= n.

A335953 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)2^k Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.