Real-rooted atlas

A114655 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(1 + x - 5\,x^{2} - 9\,x^{3} - 4\,x^{4} + n + 7\,n x + 13\,n x^{2} + 9\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 4\,x^{2} - 4\,x^{4} - 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
122
2268
32162240
423010290224
52482926083941344
62706602380353018068448
7296129070401767020256855854912
821262282174306503012402611559841586366080

All 15 rows

Technical details
Node id
n-7857580ebfc5510f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c8aa5aaa0c2f2cd0

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: 2;
   2: 2,   6;
   3: 2,  16,   22;
   4: 2,  30,  102,    90;
   5: 2,  48,  292,   608,    394;
   6: 2,  70,  660,  2380,   3530,    1806;
   7: 2,  96, 1290,  7040,  17670,   20256,    8558;
   8: 2, 126, 2282, 17430,  65030,  124026,  115598,   41586;
   9: 2, 160, 3752, 38080, 196028,  548128,  839048,  658048,  206098;
  10: 2, 198, 5832, 75768, 512316, 1954260, 4338600, 5532120, 3742290, 1037718;
%F Formula
The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (n+1+(7*n+1)*x+(13*n-5)*x^2+(9*n-9)*x^3+(2*n-4)*x^4)*P(n-1,x) + (2*x+4*x^2-4*x^4-2*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (7*n + 1)*x + (13*n - 5)*x^2 + (9*n - 9)*x^3 + (2*n - 4)*x^4)*P[n - 1, x] + (2*x + 4*x^2 - 4*x^4 - 2*x^5)*D[P[n - 1, x], x])/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n + 1 + (7*n + 1)*x + (13*n - 5)*x^2 + (9*n - 9)*x^3 + (2*n - 4)*x^4)*v[i - 1] + (2*x + 4*x^2 - 4*x^4 - 2*x^5)*deriv(v[i - 1], x))/(n + 1 + (2*n + 2)*x + (n + 1)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.