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.

1, 0, 1, 0, 1, -1, 0, 1, -1, 5, 0, 1, -41, 14, -140, 0, 1, -23, 93, -40, 100, 0, 1, -157, 2948, -3652, 7700, -15400, 0, 1, -341, 18759, -1937936, 520520, -280280, 1401400, 0, 1, -1927, 3478, -7384676, 4364360, -1430000, 5605600, -8008000

Offset: 0

Author

Peter Luschny, Sep 02 2022

Keywords

• sign
• frac
• tabl

Examples

			The rational triangle R(n, k) begins:
[0] 1;
[1] 0,     1/6;
[2] 0,    1/70,      -1/21;
[3] 0,   1/434,      -1/31,       5/93;
[4] 0,  1/2286,   -41/1905,     14/127,  -140/1143;
[5] 0, 1/11242,   -23/1533,     93/511,     -40/73,   100/219;
[6] 0, 1/53222, -157/14329, 2948/10235, -3652/2047, 7700/2047, -15400/6141;
.
Row sums are: 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, ... (A000367/A002445).
		

Links

• Peter Luschny, The P-transform.

Crossrefs

Cf. A356653 (denominators), A269941, A000367, A002445.

Programs

Maple

# Using function PTrans from A269941.
R_row := n -> seq(coeffs(p), p in PTrans(n, n -> 1/((2*n)*(2*n + 1)),
n -> (2*n)!/(2-2^(2*n)))): seq(seq(numer(r), r in R_row(n)), n = 0..8);

Formula

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.

T(n, k) = numerator([x^k] r_n(x)).