Real-rooted atlas

A003506 → Veronese 2 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
222
3639
4161228
54040585
69612030246
72243361267693
8512896448561912
911522304144028895193
1025605760432012009013930
115632140801232044005501136993
1212288337923379214784264013297428
13266247987289856465921092093613254813
1457344186368232960139776407685096182662494
Technical details
Node id
n-5b714ee1f71b153f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-702b8764f2c773f5

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:    2;
  3:    6,    3;
  4:   16,   12;
  5:   40,   40,    5;
  6:   96,  120,   30;
  7:  224,  336,  126,   7;
  8:  512,  896,  448,  56;
  9: 1152, 2304, 1440, 288, 9;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 4*x*P(n-3,x) - (4-2*x)*P(n-2,x) + 4*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 2, P(3,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 2;
P[3, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 4*x*P[n - 3, x] - (4 - 2*x)*P[n - 2, x] + 4*P[n - 1, 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] = 2; v[4] = 6 + 3*x; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - 4*x*v[i - 3] - (4 - 2*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 - 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.