A335948 -id:A335948 - cp's OEIS frontend

A335947 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 7, 0, -1, 0, 1, 0, 7, 0, -5, 0, 1, -31, 0, 7, 0, -5, 0, 1, 0, -31, 0, 49, 0, -7, 0, 1, 127, 0, -31, 0, 49, 0, -7, 0, 1, 0, 381, 0, -31, 0, 147, 0, -3, 0, 1, -2555, 0, 381, 0, -155, 0, 49, 0, -15, 0, 1

Offset: 0

Peter Luschny, Jul 01 2020

The polynomials form an Appell sequence.

The parity of n equals the parity of b(n, x). The Bernoulli polynomials do not possess this property.

			First few polynomials are:
b_0(x) = 1;
b_1(x) = x;
b_2(x) = -(1/12) + x^2;
b_3(x) = -(1/4)*x + x^3;
b_4(x) = (7/240) - (1/2)*x^2 + x^4;
b_5(x) = (7/48)*x - (5/6)*x^3 + x^5;
b_6(x) = -(31/1344) + (7/16)*x^2 - (5/4)*x^4 + x^6;
Normalized by A335949:
b_0(x) = 1;
b_1(x) = x;
b_2(x) = (-1 + 12*x^2) / 12;
b_3(x) = (-x + 4*x^3) / 4;
b_4(x) = (7 - 120*x^2 + 240*x^4) / 240;
b_5(x) = (7*x - 40*x^3 + 48*x^5) / 48;
b_6(x) = (-31 + 588*x^2 - 1680*x^4 + 1344*x^6) / 1344;
b_7(x) = (-31*x + 196*x^3 - 336*x^5 + 192*x^7) / 192;
Triangle starts:
[0] 1;
[1] 0,   1;
[2] -1,  0,   1;
[3] 0,   -1,  0,   1;
[4] 7,   0,   -1,  0,   1;
[5] 0,   7,   0,   -5,  0,  1;
[6] -31, 0,   7,   0,   -5, 0,   1;
[7] 0,   -31, 0,   49,  0,  -7,  0,  1;
[8] 127, 0,   -31, 0,   49, 0,   -7, 0,  1;
[9] 0,   381, 0,   -31, 0,  147, 0,  -3, 0, 1;

Cf. A335948 (denominators), A335949 (denominators of the polynomials).
Cf. A157779 (column 0),  A001896 (column 0 at even indices only).
Cf. A164555/A027642, A157781/A157782, A157779/A157780, A196838/A196839.

b := (n,x) -> bernoulli(n, x+1/2):
A335947row := n -> seq(numer(coeff(b(n,x), x, k)), k = 0..n):
seq(A335947row(n), n = 0..10);

b(n, 1/2) = Bernoulli(n, 1) = A164555(n)/A027642(n).
b(n, -1) = Bernoulli(n, -1/2) = A157781(n)/A157782(n).
b(n, 0) = Bernoulli(n, 1/2) = A157779(n)/A157780(n).
b(n, x) = Bernoulli(n, x + 1/2).

A335949 a(n) = denominator(b_n(x)), where b_n(x) are the polynomials defined in A335947.

Original entry on oeis.org

1, 1, 12, 4, 240, 48, 1344, 192, 3840, 1280, 33792, 3072, 5591040, 430080, 245760, 49152, 16711680, 983040, 522977280, 27525120, 1211105280, 173015040, 1447034880, 62914560, 22900899840, 4580179968, 1409286144, 469762048, 116769423360, 4026531840, 7689065201664

Offset: 0

Peter Luschny, Jul 01 2020

nonn

The sequence can also be computed without reference to the Bernoulli polynomials (ultimately thanks to the von Staudt-Clausen theorem) by the method of Kellner and Sondow (2019). Compare the SageMath program.

Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, arXiv:1902.10672 [math.NT], 2019.

Cf. A335947/A335948, A144845, A158302.

def A335949(n):
    a = set(prime_divisors(n + 1)) - set([2])
    b = (
        p
        for p in prime_range(3, (n + 2) // (2 + n % 2))
        if not p.divides(n + 1) and sum((n + 1).digits(base=p)) >= p
    )
    p = list(a.union(set(b)))
    return 4 ^ (n // 2) * mul(p)
print([A335949(n) for n in range(31)])

a(n) = min {m | m*([x^k] b(n, x)) is an integer for all k = 0..n}.
The odd part of a(n) is squarefree (A000265).
a(n) and A144845(n) have the same odd prime factors.
a(n)/A144845(n) = 4^floor(n/2)/2 for n >= 1.
a(n)/rad(a(n)) = A158302(n+1), (rad=A007947).

cp's OEIS Frontend

A335947 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

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