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
Examples
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.
Links
- Peter Luschny, Illustrating A291447
Programs
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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
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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
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.
Comments