Real-rooted atlas

A113953 → Veronese 2 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
10
2112
3448
456112
5816832
62551271104
7441006812224
877216202641560
9176544576224161536
103531173140269811713744
11660246834642184548209344
1213655578872763642035186124256
132776121122071217248695211042460928
1454172555948333460832258749652711153216
Technical details
Node id
n-8a4cd553ac75f589 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-493d6da6f1fbd65d

Observed on the stored rows ?

  • Degrees: 0, -1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • 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:   1;
  1:   0;
  2:   1,   1;
  3:   4,   4;
  4:   5,   6,   1;
  5:   8,  16,   8;
  6:  25,  51,  27,   1;
  7:  44, 100,  68,  12;
  8:  77, 216, 202,  64,  1;
  9: 176, 544, 576, 224, 16;
%F Formula
The row polynomials satisfy P(n,x) = (4+4*x)*P(n-4,x) + (4+4*x)*P(n-3,x) + (1+x)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 0, P(2,x) = 1 + x, P(3,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 0;
P[2, x_] := 1 + x;
P[3, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(4 + 4*x)*P[n - 4, x] + (4 + 4*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] = 1; v[2] = 0; v[3] = 1 + x; v[4] = 4 + 4*x; for(n = 4, N, my(i = n + 1); v[i] = (4 + 4*x)*v[i - 4] + (4 + 4*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