Real-rooted atlas

A143495 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(6 + 17\,x + 15\,x^{2} + 4\,x^{3} - n - 4\,n x - 5\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}(x) + \left(1 + 5\,x + 9\,x^{2} + 7\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2459
317432787
477295374157903
537219153670310597910041
61915123963186940696258246517119217
71048181776264041451784432211219279461071505679
8608145553812159618463680159385894538437191685834526120151759

All 15 rows

Technical details
Node id
n-b594c60e2bbf374c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9543e1a6164dd734

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:       4,        5;
   3:      17,       43,        27;
   4:      77,      295,       374,       157;
   5:     372,     1915,      3670,      3105,       979;
   6:    1915,    12396,     31869,     40696,     25824,       6517;
   7:   10481,    81776,    264041,    451784,    432211,     219279,     46107;
   8:   60814,   555381,   2159618,   4636801,   5938589,    4538437,   1916858,    345261;
   9:  372939,  3901593,  17747906,  45863285,  73660861,   75314525,  47885244,  17313789,   2726243;
  10: 2409837, 28409971, 147992153, 447196268, 864074945, 1107371875, 941501665, 512191414, 161814452, 22623525;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n-6+(4*n-17)*x+(5*n-15)*x^2+(2*n-4)*x^3)*P(n-1,x) + (1+5*x+9*x^2+7*x^3+2*x^4)*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[(-(n - 6 + (4*n - 17)*x + (5*n - 15)*x^2 + (2*n - 4)*x^3)*P[n - 1, x] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + x)]];
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] = (-(n - 6 + (4*n - 17)*x + (5*n - 15)*x^2 + (2*n - 4)*x^3)*v[i - 1] + (1 + 5*x + 9*x^2 + 7*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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.