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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.

%I A335947 #21 Jul 06 2020 07:26:12
%S A335947 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 A335947 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 A335947 -2555,0,381,0,-155,0,49,0,-15,0,1
%N 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.
%C A335947 The polynomials form an Appell sequence.
%C A335947 The parity of n equals the parity of b(n, x). The Bernoulli polynomials do not possess this property.
%F A335947 b(n, 1/2) = Bernoulli(n, 1) = A164555(n)/A027642(n).
%F A335947 b(n, -1) = Bernoulli(n, -1/2) = A157781(n)/A157782(n).
%F A335947 b(n, 0) = Bernoulli(n, 1/2) = A157779(n)/A157780(n).
%F A335947 b(n, x) = Bernoulli(n, x + 1/2).
%e A335947 First few polynomials are:
%e A335947 b_0(x) = 1;
%e A335947 b_1(x) = x;
%e A335947 b_2(x) = -(1/12) + x^2;
%e A335947 b_3(x) = -(1/4)*x + x^3;
%e A335947 b_4(x) = (7/240) - (1/2)*x^2 + x^4;
%e A335947 b_5(x) = (7/48)*x - (5/6)*x^3 + x^5;
%e A335947 b_6(x) = -(31/1344) + (7/16)*x^2 - (5/4)*x^4 + x^6;
%e A335947 Normalized by A335949:
%e A335947 b_0(x) = 1;
%e A335947 b_1(x) = x;
%e A335947 b_2(x) = (-1 + 12*x^2) / 12;
%e A335947 b_3(x) = (-x + 4*x^3) / 4;
%e A335947 b_4(x) = (7 - 120*x^2 + 240*x^4) / 240;
%e A335947 b_5(x) = (7*x - 40*x^3 + 48*x^5) / 48;
%e A335947 b_6(x) = (-31 + 588*x^2 - 1680*x^4 + 1344*x^6) / 1344;
%e A335947 b_7(x) = (-31*x + 196*x^3 - 336*x^5 + 192*x^7) / 192;
%e A335947 Triangle starts:
%e A335947 [0] 1;
%e A335947 [1] 0,   1;
%e A335947 [2] -1,  0,   1;
%e A335947 [3] 0,   -1,  0,   1;
%e A335947 [4] 7,   0,   -1,  0,   1;
%e A335947 [5] 0,   7,   0,   -5,  0,  1;
%e A335947 [6] -31, 0,   7,   0,   -5, 0,   1;
%e A335947 [7] 0,   -31, 0,   49,  0,  -7,  0,  1;
%e A335947 [8] 127, 0,   -31, 0,   49, 0,   -7, 0,  1;
%e A335947 [9] 0,   381, 0,   -31, 0,  147, 0,  -3, 0, 1;
%p A335947 b := (n,x) -> bernoulli(n, x+1/2):
%p A335947 A335947row := n -> seq(numer(coeff(b(n,x), x, k)), k = 0..n):
%p A335947 seq(A335947row(n), n = 0..10);
%Y A335947 Cf. A335948 (denominators), A335949 (denominators of the polynomials).
%Y A335947 Cf. A157779 (column 0),  A001896 (column 0 at even indices only).
%Y A335947 Cf. A164555/A027642, A157781/A157782, A157779/A157780, A196838/A196839.
%K A335947 sign,frac,tabl
%O A335947 0,11
%A A335947 _Peter Luschny_, Jul 01 2020