Real-rooted atlas

A013609 → basis A008288 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-9 + 12\,x - 4\,x^{2}\right)\,P_{n-4}(x) + \left(24 - 4\,x - 8\,x^{2}\right)\,P_{n-3}(x) + \left(-22 - 12\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)
\(P_{2}(x) = 9 + 32\,x + 12\,x^{2}\)
\(P_{3}(x) = 27 + 164\,x + 156\,x^{2} + 32\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
29321253
32716415632379
4817041176576802617
5243275668405760184019217631
6729102083421242688229605376448116621
721873645215522026294420744080448147841024760499

All 15 rows

Technical details
Node id
n-39f9b82cbc6d1b43 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c9b7a2e56f9283c2

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:     3,      4;
  2:     9,     32,      12;
  3:    27,    164,     156,      32;
  4:    81,    704,    1176,     576,      80;
  5:   243,   2756,    6840,    5760,    1840,     192;
  6:   729,  10208,   34212,   42688,   22960,    5376,     448;
  7:  2187,  36452,  155220,  262944,  207440,   80448,   14784,   1024;
  8:  6561, 126848,  658224, 1428864, 1524960,  858624,  258048,  38912,  2304;
  9: 19683, 433028, 2657136, 7095808, 9707040, 7350912, 3179008, 776192, 99072, 5120;
%F Formula
The row polynomials satisfy P(n,x) = -(9-12*x+4*x^2)*P(n-4,x) + (24-4*x-8*x^2)*P(n-3,x) - (22+12*x+4*x^2)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 9 + 32*x + 12*x^2, P(3,x) = 27 + 164*x + 156*x^2 + 32*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 9 + 32*x + 12*x^2;
P[3, x_] := 27 + 164*x + 156*x^2 + 32*x^3;
P[n_, x_] := P[n, x] = Expand[-(9 - 12*x + 4*x^2)*P[n - 4, x] + (24 - 4*x - 8*x^2)*P[n - 3, x] - (22 + 12*x + 4*x^2)*P[n - 2, x] + (8 + 4*x)*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] = 1; v[2] = 3 + 4*x; v[3] = 9 + 32*x + 12*x^2; v[4] = 27 + 164*x + 156*x^2 + 32*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9 - 12*x + 4*x^2)*v[i - 4] + (24 - 4*x - 8*x^2)*v[i - 3] - (22 + 12*x + 4*x^2)*v[i - 2] + (8 + 4*x)*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

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.