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

Author

Peter Luschny, Jul 01 2020

Keywords

• sign
• frac
• tabl

Comments

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.

Examples

			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;
		

Crossrefs

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.

Programs

Maple

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);

Formula

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).