Real-rooted atlas

A039756 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 5\,x + 6\,x^{2} + 2\,x^{3} - 2\,n x - 4\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
216613
3116382479
4144184256116601
51126824189618446485339
613703606122641923614296408853861
711100157987493616880419908811916028640607527
8132887017644816013612562265696212896010631682199207560625
919850317984267003210512872231635683038060823642944101071521832224102637235
1012953414717021598875279483944222758864382473456412380480273253264101954912164301761506225085
1118858469450149667263259587575220675370884463865328625295929657093226403288406144108720905615755404823726435583
1212657323330770459159657644644904601881264284849687988736868394056961024435059688096791404841189952640122142996481606879040398852249097
1317971741617553203669022072335982943001695033902005368867205121138590063040166632017080016944313696321178991754368536540555136144126757696173502557447120170905995
14123914987928330942307593720825479776679615223704963605696744508456143667967024002536538906859231861676632928284288015693121764884988096072600295189121781336869248197553645440134408217821205
Technical details
Node id
n-5a02758adff280aa · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-078627b5c58f7a3e

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,    2;
  2: 1,    6,      6;
  3: 1,   16,     38,      24;
  4: 1,   44,    184,     256,      116;
  5: 1,  126,    824,    1896,     1844,      648;
  6: 1,  370,   3606,   12264,    19236,    14296,     4088;
  7: 1, 1100,  15798,   74936,   168804,   199088,   119160,    28640;
  8: 1, 3288,  70176,  448160,  1361256,  2265696,  2128960,  1063168,   219920;
  9: 1, 9850, 317984, 2670032, 10512872, 23163568, 30380608, 23642944, 10107152, 1832224;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1-(2*n-5)*x-(4*n-6)*x^2-(2*n-2)*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx 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[((1 - (2*n - 5)*x - (4*n - 6)*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], 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] = ((1 - (2*n - 5)*x - (4*n - 6)*x^2 - (2*n - 2)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Candidate equalities ?

Same fingerprint ?

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.