cp's OEIS Frontend

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).

%I A291447 #15 Aug 26 2017 08:21:35
%S A291447 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,
%T A291447 1968,-216,64,0,0,0,1,-15,240,-1580,37140,-9900,10400,-5760,14400,0,0,
%U A291447 0,1,-31,4924,-11680,488640,-238680,496320,-639360,5486400,-216000,518400
%N 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).
%C A291447 The Bernoulli median numbers are A212196/A181131. See A290694 for further comments.
%H A291447 Peter Luschny, <a href="/A291447/a291447.jpg">Illustrating A291447</a>
%F A291447 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.
%e A291447 Triangle starts:
%e A291447 [0, 1]
%e A291447 [0, 0, 0, 1]
%e A291447 [0, 0, 0, 1, -1, 4]
%e A291447 [0, 0, 0, 1, -3, 48, -12, 36]
%e A291447 [0, 0, 0, 1, -7, 268, -176, 1968, -216, 64]
%e A291447 [0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400]
%e A291447 The first few polynomials are:
%e A291447 P_0(x) = x.
%e A291447 P_1(x) = (1/3)*x^3.
%e A291447 P_2(x) = (4/5)*x^5 - x^4 + (1/3)*x^3.
%e A291447 P_3(x) = (36/7)*x^7 - 12*x^6 + (48/5)*x^5 - 3*x^4 + (1/3)*x^3.
%e A291447 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.
%e A291447 Evaluated at x = 1 this gives a decomposition of the Bernoulli median numbers:
%e A291447 BM(0) = 1     =    1.
%e A291447 BM(1) = 1/3   =  1/3.
%e A291447 BM(2) = 2/15  =  4/5 -   1 +    1/3.
%e A291447 BM(3) = 8/105 = 36/7 -  12 +   48/5 -   3 +   1/3.
%e A291447 BM(4) = 8/105 =   64 - 216 + 1968/7 - 176 + 268/5 - 7 + 1/3.
%p A291447 # The function BG_row is defined in A290694.
%p A291447 seq(BG_row(2, n, "num", "val"), n=0..12);        # A212196
%p A291447 seq(BG_row(2, n, "den", "val"), n=0..12);        # A181131
%p A291447 seq(print(BG_row(2, n, "num", "poly")), n=0..7); # A291447 (this seq.)
%p A291447 seq(print(BG_row(2, n, "den", "poly")), n=0..9); # A291448
%t A291447 T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j, 0, n}]^2, x];
%t A291447 Trow[n_] := CoefficientList[T[n], x] // Numerator;
%t A291447 Table[Trow[r], {r, 0, 6}] // Flatten
%Y A291447 Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.
%K A291447 sign,tabf,frac
%O A291447 0,12
%A A291447 _Peter Luschny_, Aug 24 2017