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A356652 Triangle read by rows. Numerators of the coefficients of a sequence of rational polynomials r_n(x) with r_n(1) = B(2*n), where B(n) are the Bernoulli numbers.

%I A356652 #10 Sep 02 2022 08:00:39
%S A356652 1,0,1,0,1,-1,0,1,-1,5,0,1,-41,14,-140,0,1,-23,93,-40,100,0,1,-157,
%T A356652 2948,-3652,7700,-15400,0,1,-341,18759,-1937936,520520,-280280,
%U A356652 1401400,0,1,-1927,3478,-7384676,4364360,-1430000,5605600,-8008000
%N A356652 Triangle read by rows. Numerators of the coefficients of a sequence of rational polynomials r_n(x) with r_n(1) = B(2*n), where B(n) are the Bernoulli numbers.
%H A356652 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.
%F A356652 Let r_n(x) = ((2*n)! / (2-2^(2*n))) * Sum_{p in P_n} (-x)^(p_1) * binomial(p_1, p_2) * binomial(p_2, p_3) * ... * binomial(p_{n-1}, p_{n}) * (2*3)^(-p_1) * (4*5)^(-p_2) * ... * (2*n*(2*n+1))^(-p_n), where P_n are the partitions of n and we say that p is a partition of n if and only if p = (p_{1}, ..., p_{n}), the p_{i} are integers, Sum_{1<=i<=n} p_i = n and p_{1} >= p_{2} >= ... >= p_{n} >= 0.
%F A356652 T(n, k) = numerator([x^k] r_n(x)).
%e A356652 The rational triangle R(n, k) begins:
%e A356652 [0] 1;
%e A356652 [1] 0,     1/6;
%e A356652 [2] 0,    1/70,      -1/21;
%e A356652 [3] 0,   1/434,      -1/31,       5/93;
%e A356652 [4] 0,  1/2286,   -41/1905,     14/127,  -140/1143;
%e A356652 [5] 0, 1/11242,   -23/1533,     93/511,     -40/73,   100/219;
%e A356652 [6] 0, 1/53222, -157/14329, 2948/10235, -3652/2047, 7700/2047, -15400/6141;
%e A356652 .
%e A356652 Row sums are: 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, ... (A000367/A002445).
%p A356652 # Using function PTrans from A269941.
%p A356652 R_row := n -> seq(coeffs(p), p in PTrans(n, n -> 1/((2*n)*(2*n + 1)),
%p A356652 n -> (2*n)!/(2-2^(2*n)))): seq(seq(numer(r), r in R_row(n)), n = 0..8);
%Y A356652 Cf. A356653 (denominators), A269941, A000367, A002445.
%K A356652 sign,frac,tabl
%O A356652 0,10
%A A356652 _Peter Luschny_, Sep 02 2022