A335947 - OEIS

%I #21 Jul 06 2020 07:26:12

%S 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,

%T 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,

%U -2555,0,381,0,-155,0,49,0,-15,0,1

%N 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.

%C The polynomials form an Appell sequence.

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

%F b(n, 1/2) = Bernoulli(n, 1) = A164555(n)/A027642(n).

%F b(n, -1) = Bernoulli(n, -1/2) = A157781(n)/A157782(n).

%F b(n, 0) = Bernoulli(n, 1/2) = A157779(n)/A157780(n).

%F b(n, x) = Bernoulli(n, x + 1/2).

%e First few polynomials are:

%e b_0(x) = 1;

%e b_1(x) = x;

%e b_2(x) = -(1/12) + x^2;

%e b_3(x) = -(1/4)*x + x^3;

%e b_4(x) = (7/240) - (1/2)*x^2 + x^4;

%e b_5(x) = (7/48)*x - (5/6)*x^3 + x^5;

%e b_6(x) = -(31/1344) + (7/16)*x^2 - (5/4)*x^4 + x^6;

%e Normalized by A335949:

%e b_0(x) = 1;

%e b_1(x) = x;

%e b_2(x) = (-1 + 12*x^2) / 12;

%e b_3(x) = (-x + 4*x^3) / 4;

%e b_4(x) = (7 - 120*x^2 + 240*x^4) / 240;

%e b_5(x) = (7*x - 40*x^3 + 48*x^5) / 48;

%e b_6(x) = (-31 + 588*x^2 - 1680*x^4 + 1344*x^6) / 1344;

%e b_7(x) = (-31*x + 196*x^3 - 336*x^5 + 192*x^7) / 192;

%e Triangle starts:

%e [0] 1;

%e [1] 0,   1;

%e [2] -1,  0,   1;

%e [3] 0,   -1,  0,   1;

%e [4] 7,   0,   -1,  0,   1;

%e [5] 0,   7,   0,   -5,  0,  1;

%e [6] -31, 0,   7,   0,   -5, 0,   1;

%e [7] 0,   -31, 0,   49,  0,  -7,  0,  1;

%e [8] 127, 0,   -31, 0,   49, 0,   -7, 0,  1;

%e [9] 0,   381, 0,   -31, 0,  147, 0,  -3, 0, 1;

%p b := (n,x) -> bernoulli(n, x+1/2):

%p A335947row := n -> seq(numer(coeff(b(n,x), x, k)), k = 0..n):

%p seq(A335947row(n), n = 0..10);

%Y Cf. A335948 (denominators), A335949 (denominators of the polynomials).

%Y Cf. A157779 (column 0),  A001896 (column 0 at even indices only).

%Y Cf. A164555/A027642, A157781/A157782, A157779/A157780, A196838/A196839.

%K sign,frac,tabl

%O 0,11

%A _Peter Luschny_, Jul 01 2020