Real-rooted atlas

A111577 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 9\,x + 14\,x^{2} + 6\,x^{3} - 3\,n x - 6\,n x^{2} - 3\,n x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
317715
41245735117
51893664912141161
61346219448414514152313419
7113711292142064631414497612349176823
815468767193483917499098461684854491120522624157
91218534659282848453824964113107541117821985640769112084943237233
10187390291182423366337874648171834931432297263981710798217014341712219767780492915
11134953518722331193712298912329958242783140539702950224088828724259440480092897314114394299215279390207
1211398112123366517162812823994738305303112802439063869328984852991210297446018187241099885432130491056181819256321321951627045
13155924178286127861389537520798663233824392656498794983438785176163562286853818417549433591393030431572679678002246193679465244245266543817257780463545
141223696345643701790120539721017103543765282049206965118751474679331139229704794203514414269583991114025507826297226829567380758117194559925443027161086429350974470746174143123064603
15189478499388124538371063499358596109841926282676165426307283021778738070718352066196288091687451527145611310481524380901858946485432873575324806143482662104030775548274967288089959153084700412994427371496257767
Technical details
Node id
n-dccb22bb1a700e42 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ab299508b34125dc

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,     2;
   3: 1,     7,       7;
   4: 1,    24,      57,       35;
   5: 1,    89,     366,      491,      214;
   6: 1,   346,    2194,     4841,     4514,      1523;
   7: 1,  1371,   12921,    42064,    63141,     44976,     12349;
   8: 1,  5468,   76719,   348391,   749909,    846168,    485449,    112052;
   9: 1, 21853,  465928,  2848453,  8249641,  13107541,  11782198,   5640769,  1120849;
  10: 1, 87390, 2911824, 23366337, 87464817, 183493143, 229726398, 171079821, 70143417, 12219767;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1-(3*n-9)*x-(6*n-14)*x^2-(3*n-6)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (3*n - 9)*x - (6*n - 14)*x^2 - (3*n - 6)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 - (3*n - 9)*x - (6*n - 14)*x^2 - (3*n - 6)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.