The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A290694 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) = Bernoulli(n, 1). 6
 0, 1, 0, 0, 1, 0, 0, -1, 2, 0, 0, 1, -2, 3, 0, 0, -1, 14, -9, 24, 0, 0, 1, -10, 75, -48, 20, 0, 0, -1, 62, -135, 312, -300, 720, 0, 0, 1, -42, 903, -1680, 2800, -2160, 630, 0, 0, -1, 254, -1449, 40824, -21000, 27360, -17640, 4480 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Consider a family of integrals I_m(n) = Integral_{x=0..1} P'(n, x)^m with P'(n,x) = Sum_{k=0..n}(-1)^(n-k)*Stirling2(n, k)*k!*x^k (see A278075 for the coefficients). I_1(n) are the Bernoulli numbers A164555/A027642, I_2(n) are the Bernoulli median numbers A212196/A181131, I_3(n) are the numbers A291449/A291450. The coefficients of the polynomials P_n(x)^m are for m = 1 A290694/A290695 and for m = 2 A291447/A291448. Only omega(Clausen(n)) = A001221(A160014(n,1)) = A067513(n) coefficients are rational numbers if n is even. For odd n > 1 there are two rational coefficients. Let C_k(n) = [x^k] P_n(x), k > 0 and n even. Conjecture: k is a prime factor of Clausen(n) <=> k = denominator(C_k(n)) <=> k does not divide Stirling2(n, k-1)*(k-1)!. (Note that by a comment in A019538 Stirling2(n, k-1)*(k-1)! is the number of chain topologies on an n-set having k open sets.) LINKS Table of n, a(n) for n=0..53. Peter Luschny, Illustrating A290694. FORMULA T(n, k) = Numerator(Stirling2(n, k - 1)*(k - 1)!/k) if k > 0 else 0; for n >= 0 and 0 <= k <= n+1. EXAMPLE Triangle starts: [0, 1] [0, 0, 1] [0, 0, -1, 2] [0, 0, 1, -2, 3] [0, 0, -1, 14, -9, 24] [0, 0, 1, -10, 75, -48, 20] [0, 0, -1, 62, -135, 312, -300, 720] The first few polynomials are: P_0(x) = x. P_1(x) = (1/2)*x^2. P_2(x) = -(1/2)*x^2 + (2/3)*x^3. P_3(x) = (1/2)*x^2 - 2*x^3 + (3/2)*x^4. P_4(x) = -(1/2)*x^2 + (14/3)*x^3 - 9*x^4 + (24/5)*x^5. P_5(x) = (1/2)*x^2 - 10*x^3 + (75/2)*x^4 - 48*x^5 + 20*x^6. P_6(x) = -(1/2)*x^2 + (62/3)*x^3 - 135*x^4 + 312*x^5 - 300*x^6 + (720/7)*x^7. Evaluated at x = 1 this gives an additive decomposition of the Bernoulli numbers: B(0) = 1 = 1. B(1) = 1/2 = 1/2. B(2) = 1/6 = -1/2 + 2/3. B(3) = 0 = 1/2 - 2 + 3/2. B(4) = -1/30 = -1/2 + 14/3 - 9 + 24/5. B(5) = 0 = 1/2 - 10 + 75/2 - 48 + 20. B(6) = 1/42 = -1/2 + 62/3 - 135 + 312 - 300 + 720/7. MAPLE BG_row := proc(m, n, frac, val) local F, g, v; F := (n, x) -> add((-1)^(n-k)*Stirling2(n, k)*k!*x^k, k=0..n): g := x -> int(F(n, x)^m, x): `if`(val = "val", subs(x=1, g(x)), [seq(coeff(g(x), x, j), j=0..m*n+1)]): `if`(frac = "num", numer(%), denom(%)) end: seq(BG_row(1, n, "num", "val"), n=0..16); # A164555 seq(BG_row(1, n, "den", "val"), n=0..16); # A027642 seq(print(BG_row(1, n, "num", "poly")), n=0..12); # A290694 (this seq.) seq(print(BG_row(1, n, "den", "poly")), n=0..12); # A290695 # Alternatively: T_row := n -> numer(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 6 do T_row(n) od; MATHEMATICA T[n_, k_] := If[k > 0, Numerator[StirlingS2[n, k - 1]*(k - 1)! / k], 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n+1}] // Flatten CROSSREFS Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448. Cf. A160014, A278075. Sequence in context: A167948 A111755 A144528 * A146164 A263141 A051510 Adjacent sequences: A290691 A290692 A290693 * A290695 A290696 A290697 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 April 21 11:18 EDT 2024. Contains 371853 sequences. (Running on oeis4.)