A291447 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = BernoulliMedian(n).

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, -1, 4, 0, 0, 0, 1, -3, 48, -12, 36, 0, 0, 0, 1, -7, 268, -176, 1968, -216, 64, 0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400, 0, 0, 0, 1, -31, 4924, -11680, 488640, -238680, 496320, -639360, 5486400, -216000, 518400

Offset

0,12

Comments

The Bernoulli median numbers are A212196/A181131. See A290694 for further comments.

Links

Table of n, a(n) for n=0..55.

Peter Luschny, Illustrating A291447

Formula

T(n,k) = Numerator([x^k] Integral(Sum_{j=0..n}(-1)^(n-j)*Stirling2(n,j)*j!*x^j)^m) for m = 2, n >= 0 and k = 0..m*n+1.

Example

Triangle starts:
[0, 1]
[0, 0, 0, 1]
[0, 0, 0, 1, -1, 4]
[0, 0, 0, 1, -3, 48, -12, 36]
[0, 0, 0, 1, -7, 268, -176, 1968, -216, 64]
[0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400]
The first few polynomials are:
P_0(x) = x.
P_1(x) = (1/3)*x^3.
P_2(x) = (4/5)*x^5 - x^4 + (1/3)*x^3.
P_3(x) = (36/7)*x^7 - 12*x^6 + (48/5)*x^5 - 3*x^4 + (1/3)*x^3.
P_4(x) = 64*x^9 - 216*x^8 + (1968/7)*x^7 - 176*x^6 + (268/5)*x^5 - 7*x^4 +(1/3)*x^3.
Evaluated at x = 1 this gives a decomposition of the Bernoulli median numbers:
BM(0) = 1     =    1.
BM(1) = 1/3   =  1/3.
BM(2) = 2/15  =  4/5 -   1 +    1/3.
BM(3) = 8/105 = 36/7 -  12 +   48/5 -   3 +   1/3.
BM(4) = 8/105 =   64 - 216 + 1968/7 - 176 + 268/5 - 7 + 1/3.

Maple

# The function BG_row is defined in A290694.
seq(BG_row(2, n, "num", "val"), n=0..12);        # A212196
seq(BG_row(2, n, "den", "val"), n=0..12);        # A181131
seq(print(BG_row(2, n, "num", "poly")), n=0..7); # A291447 (this seq.)
seq(print(BG_row(2, n, "den", "poly")), n=0..9); # A291448

Mathematica

T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j, 0, n}]^2, x];
Trow[n_] := CoefficientList[T[n], x] // Numerator;
Table[Trow[r], {r, 0, 6}] // Flatten

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Sequence in context: A245817 A343316 A277115 * A152894 A152898 A368661

Adjacent sequences: A291444 A291445 A291446 * A291448 A291449 A291450

Keyword

sign,tabf,frac

Author

Peter Luschny, Aug 24 2017

Status

approved