The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). 7
 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 (list; graph; refs; listen; history; text; internal format)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 18 15:24 EDT 2024. Contains 372664 sequences. (Running on oeis4.)