Real-rooted atlas

A075497 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
2181120
31155549120
4124166384257832
51353901690284115396496
6148785548017011225041029956128
71631421145957261117446119086375905529920
8180238033824247366944720184775617280646094415413632
91993756706447156384003986123291642033067616640785528445159379200
10112056551360801827882141550086352939016332504023293050116981108049134923694855680
111143819524568542317224347835826947907810036984102211090981278027482718299123274870269298629446656
1211681150642064090472471193485929808985565028666720159465642713070968093634558382002207576878725120905441113231613952
131195157306889741811080729881780531545805722141302534893090014303256408555117438518024594446924447502247142872809568780920671563847245824
141224210211086904343032696929242329158113393798219799524584495225311723535045232418991227391164448551097846006722152887308029024612066492002181922658392801280
Technical details
Node id
n-6a07454e7661ed77 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6bc75e460659aa09

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,  3;
  2: 1,  8,   11;
  3: 1, 15,   55,    49;
  4: 1, 24,  166,   384,    257;
  5: 1, 35,  390,  1690,   2841,    1539;
  6: 1, 48,  785,  5480,  17011,   22504,    10299;
  7: 1, 63, 1421, 14595,  72611,  174461,   190863,    75905;
  8: 1, 80, 2380, 33824, 247366,  944720,  1847756,  1728064,   609441;
  9: 1, 99, 3756, 70644, 715638, 4003986, 12329164, 20330676, 16640785, 5284451;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(2*n+3)*x+(6*n+1)*x^2+(6*n-3)*x^3+(2*n-2)*x^4)*P(n-1,x) - (2*x^2+6*x^3+6*x^4+2*x^5)*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 + 3)*x + (6*n + 1)*x^2 + (6*n - 3)*x^3 + (2*n - 2)*x^4)*P[n - 1, x] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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 + 3)*x + (6*n + 1)*x^2 + (6*n - 3)*x^3 + (2*n - 2)*x^4)*v[i - 1] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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.