Real-rooted atlas

A132774 → Veronese 2 → x ↦ x+1

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-1 - 2\,x - x^{2}\right)\,P_{n-4}(x) + \left(2 + 2\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 5 + 5\,x\)
\(P_{3}(x) = 6 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
25510
36612
4918936
510201040
613393913104
714424214112
817681026817272
918721087218288
102110521021010521672
112211022022011022704
1225150375500375150251600
1326156390520390156261664
142920360910151015609203293712
Technical details
Node id
n-6b20e3ef501a5905 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6052c7b42ebfb2ec

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  2;
  2:  5,  5;
  3:  6,  6;
  4:  9, 18,   9;
  5: 10, 20,  10;
  6: 13, 39,  39, 13;
  7: 14, 42,  42, 14;
  8: 17, 68, 102, 68, 17;
  9: 18, 72, 108, 72, 18;
%F Formula
The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-4,x) + (2+2*x)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 5 + 5*x, P(3,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 5 + 5*x;
P[3, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 4, x] + (2 + 2*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] = 2; v[3] = 5 + 5*x; v[4] = 6 + 6*x; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 4] + (2 + 2*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

Same fingerprint ?

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. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [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
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.