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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. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 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. LINKS 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). EXAMPLE 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; 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); 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. Sequence in context: A340906 A136115 A061846 * A293530 A199603 A121570 Adjacent sequences: A335944 A335945 A335946 * A335948 A335949 A335950 KEYWORD sign,frac,tabl AUTHOR Peter Luschny, Jul 01 2020 STATUS approved

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Last modified January 28 12:23 EST 2023. Contains 359873 sequences. (Running on oeis4.)