Real-rooted atlas

A078812 → Veronese 2 (j=1) → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-4}(x) + \left(4\right)\,P_{n-3}(x) + \left(-5 + x\right)\,P_{n-2}(x) + \left(4\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 4\)
\(P_{4}(x) = 11 + x\)

Rows ?

nx0x1x2x3Pn(1)
10
211
344
411112
528836
672381111
718814412344
84934898111064

All 15 rows

Technical details
Node id
n-93e1012722d95429 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1da4f95bfa2fb1e2

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half affine: slope 1, intercept -2)
  • 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:    0;
   2:    1;
   3:    4;
   4:   11,    1;
   5:   28,    8;
   6:   72,   38,    1;
   7:  188,  144,   12;
   8:  493,  489,   81,   1;
   9: 1292, 1568,  412,  16;
  10: 3383, 4858, 1779, 140, 1;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-4,x) + 4*P(n-3,x) - (5-x)*P(n-2,x) + 4*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 4, P(4,x) = 11 + x.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 4;
P[4, x_] := 11 + x;
P[n_, x_] := P[n, x] = Expand[-P[n - 4, x] + 4*P[n - 3, x] - (5 - x)*P[n - 2, x] + 4*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 4; v[4] = 11 + x; for(n = 5, N, my(i = n); v[i] = -v[i - 4] + 4*v[i - 3] - (5 - x)*v[i - 2] + 4*v[i - 1]); 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  4. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  5. [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
  6. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023