Real-rooted atlas

A039755 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 - 7\,x^{2} - 10\,x^{3} - 4\,x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 6\,x^{3} - 6\,x^{4} - 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
316613
4112342471
5120112208116457
6130280100013966483355
71425903480907610232408827509
81561106980041636855688132028640248127
91721904237441506964970568445446961922199202434129
101903072514084584722241456602656087457926382864183222425741939
1111104710102000122119283600163302584074889920950169766234684816430176291397789
121132693018876029285522688470414800033648874848095834305610817882246458854721575540483510328695
13115698563300006450972767616965653859202587368960732583121612658520512128866731527068248832160687904044782460313
1411821362455026413248092198819192189678860811577435584451588897761117905222721727614869761603394108168142656697617350255744602513988107
15121018382881608256536284747863125713999720451071603842337980284647926264265921742255607456243675548480020800319537289844141669121975536454408518757813637
Technical details
Node id
n-1fc15c499dc7e8e8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f9cef97b53c8c582

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,  6,    6;
   4: 1, 12,   34,    24;
   5: 1, 20,  112,   208,    116;
   6: 1, 30,  280,  1000,   1396,     648;
   7: 1, 42,  590,  3480,   9076,   10232,    4088;
   8: 1, 56, 1106,  9800,  41636,   85568,   81320,   28640;
   9: 1, 72, 1904, 23744, 150696,  497056,  844544,  696192,  219920;
  10: 1, 90, 3072, 51408, 458472, 2241456, 6026560, 8745792, 6382864, 1832224;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+2*n*x+(6*n-7)*x^2+(6*n-10)*x^3+(2*n-4)*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 >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 2*n*x + (6*n - 7)*x^2 + (6*n - 10)*x^3 + (2*n - 4)*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, 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 + 2*n*x + (6*n - 7)*x^2 + (6*n - 10)*x^3 + (2*n - 4)*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, v[i])))};
T(10)

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.