Real-rooted atlas

A143495 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2145
3191727
41156477157
5122157427372979
61303151435282119156517
7139560376012376188901048146107
814991784354093610442612968360814345261
91601414169821130014247188790479180813729392726243
101722082315422743231413636430407774665176721438240983722623525
1118529555502060431740717951698478043249600644327715095006116360786196712171
12199407091245123317710500105572568371994868154351190915672954873999766331163932051787356765
1311145467145145236474724795012170890126777798978231729779644110763845109106828325086734486554945316929897395
1411307189222937430630254478281462553234266026532210342829354268384536424525583270047138737464273380998296713065156166808851541
151147928233233275064991126895771155049324818417714440183355212135885509760311677608606471283380932445938022713237681039229541903604531706299041211
Technical details
Node id
n-753eebbdd6abb3d8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-adf2adbefbd3f634

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,  4;
   3: 1,  9,   17;
   4: 1, 15,   64,    77;
   5: 1, 22,  157,   427,    372;
   6: 1, 30,  315,  1435,   2821,    1915;
   7: 1, 39,  560,  3760,  12376,   18890,   10481;
   8: 1, 49,  917,  8435,  40936,  104426,  129683,   60814;
   9: 1, 60, 1414, 16982, 113001,  424718,  879047,  918081,  372939;
  10: 1, 72, 2082, 31542, 274323, 1413636, 4304077, 7466517, 6721438, 2409837;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(n+4)*x+(3*n+3)*x^2+(3*n-2)*x^3+(n-2)*x^4)*P(n-1,x) - (x^2+3*x^3+3*x^4+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 + (n + 4)*x + (3*n + 3)*x^2 + (3*n - 2)*x^3 + (n - 2)*x^4)*P[n - 1, x] - (x^2 + 3*x^3 + 3*x^4 + 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 + (n + 4)*x + (3*n + 3)*x^2 + (3*n - 2)*x^3 + (n - 2)*x^4)*v[i - 1] - (x^2 + 3*x^3 + 3*x^4 + 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.