A157782
Denominator of Bernoulli(n, -1/2).
Original entry on oeis.org
1, 1, 12, 4, 240, 16, 1344, 64, 3840, 256, 33792, 1024, 5591040, 4096, 49152, 16384, 16711680, 65536, 104595456, 262144, 173015040, 1048576, 289406976, 4194304, 22900899840, 16777216, 201326592, 67108864, 116769423360, 268435456
Offset: 0
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Table[Denominator[BernoulliB[n, -1/2]], {n, 0, 50}] (* Vincenzo Librandi, Mar 19 2014 *)
A285866
a(n) = numerator((-2)^n*Sum_{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
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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
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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 *)
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# 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
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.
Original entry on oeis.org
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
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).
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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);
Showing 1-3 of 3 results.
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