Real-rooted atlas

A113214 → Veronese 2 (j=1) → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
233
3314
45510
51011122
61421742
7234523192
83987579192
961159136391396
109929730812111836
111625406513315911744
122609621339845221133640
134211714269620156948317624
1468330225288457719783651515928
Technical details
Node id
n-2d6681969c81afb0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e2879f044171a7f8

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  1;
  2:  3;
  3:  3,   1;
  4:  5,   5;
  5: 10,  11,   1;
  6: 14,  21,   7;
  7: 23,  45,  23,  1;
  8: 39,  87,  57,  9;
  9: 61, 159, 136, 39, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-4,x) + (2+2*x)*P(n-3,x) + (1+x)*P(n-2,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3, P(3,x) = 3 + x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 3;
P[3, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 4, x] + (2 + 2*x)*P[n - 3, x] + (1 + x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 3; v[4] = 3 + x; for(n = 4, N, my(i = n + 1); v[i] = (1 + x)*v[i - 4] + (2 + 2*x)*v[i - 3] + (1 + x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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. [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
  5. [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