Real-rooted atlas

A008955 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + n^{2} + 3\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
210332770
31005208975132030
4170012140324093832216929101500
5442004023401461774264923123945768633797815500
61635400181131808345645820473104928199317520674907163026667859705000
78177000010675636005966027720184987417923436803260138254777875236194913796239640033128096045000

All 15 rows

Technical details
Node id
n-e1501c7256f70865 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b84972725564956e

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 0)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:            2,             3;
  2:           10,            33,             27;
  3:          100,           520,            897,             513;
  4:         1700,         12140,          32409,           38322,           16929;
  5:        44200,        402340,        1461774,         2649231,         2394576,           863379;
  6:      1635400,      18113180,       83456458,       204731049,       281993175,        206749071,         63026667;
  7:     81770000,    1067563600,     5966027720,     18498741792,     34368032601,      38254777875,      23619491379,       6239640033;
  8:   5315050000,   79939964000,   525507506200,   1972035792360,   4620259810233,    6920036767404,    6470133285510,    3452490990036,    804913564257;
  9: 435834100000, 7421430198000, 56121829640400, 247364658484120, 700303138593786, 1320545363995107, 1658516922498672, 1337735986721082, 628758943644942, 131200910973891;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n^2+1+(3*n^2+2)*x+(2*n^2+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 1 + (3*n^2 + 2)*x + (2*n^2 + 1)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 1 + (3*n^2 + 2)*x + (2*n^2 + 1)*x^2)*v[i - 1])/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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.