Real-rooted atlas

A013609 → basis A008288 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 24\,x + 36\,x^{2}\right)\,P_{n-4}(x) + \left(-24\,x^{2} + 16\,x^{3}\right)\,P_{n-4}'(x) + \left(24 - 8\,x + 72\,x^{2}\right)\,P_{n-3}(x) + \left(8\,x^{2} + 32\,x^{3}\right)\,P_{n-3}'(x) + \left(-22 - 24\,x + 36\,x^{2}\right)\,P_{n-2}(x) + \left(24\,x^{2} + 16\,x^{3}\right)\,P_{n-2}'(x) + \left(8 + 8\,x\right)\,P_{n-1}(x) + \left(-8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 8\,x\)
\(P_{2}(x) = 9 + 64\,x + 36\,x^{2}\)
\(P_{3}(x) = 27 + 328\,x + 468\,x^{2} + 128\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
296436109
327328468128951
4811408352823044007721
5243551220520230409200115259667
672920416102636170752114800322563136444725
72187729044656601051776103720048268810348881923224095

All 15 rows

Technical details
Node id
n-4d47dd091803573d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e250454d15f5415f

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,      8;
  2:     9,     64,      36;
  3:    27,    328,     468,      128;
  4:    81,   1408,    3528,     2304,      400;
  5:   243,   5512,   20520,    23040,     9200,     1152;
  6:   729,  20416,  102636,   170752,   114800,    32256,     3136;
  7:  2187,  72904,  465660,  1051776,  1037200,   482688,   103488,    8192;
  8:  6561, 253696, 1974672,  5715456,  7624800,  5151744,  1806336,  311296,  20736;
  9: 19683, 866056, 7971408, 28383232, 48535200, 44105472, 22253056, 6209536, 891648, 51200;
%F Formula
The row polynomials satisfy P(n,x) = -(9-24*x-36*x^2)*P(n-4,x) - (24*x^2-16*x^3)*d/dx P(n-4,x) + (24-8*x+72*x^2)*P(n-3,x) + (8*x^2+32*x^3)*d/dx P(n-3,x) - (22+24*x-36*x^2)*P(n-2,x) + (24*x^2+16*x^3)*d/dx P(n-2,x) + (8+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 8*x, P(2,x) = 9 + 64*x + 36*x^2, P(3,x) = 27 + 328*x + 468*x^2 + 128*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[2, x_] := 9 + 64*x + 36*x^2;
P[3, x_] := 27 + 328*x + 468*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[-(9 - 24*x - 36*x^2)*P[n - 4, x] - (24*x^2 - 16*x^3)*D[P[n - 4, x], x] + (24 - 8*x + 72*x^2)*P[n - 3, x] + (8*x^2 + 32*x^3)*D[P[n - 3, x], x] - (22 + 24*x - 36*x^2)*P[n - 2, x] + (24*x^2 + 16*x^3)*D[P[n - 2, x], x] + (8 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], 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 + 8*x; v[3] = 9 + 64*x + 36*x^2; v[4] = 27 + 328*x + 468*x^2 + 128*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9 - 24*x - 36*x^2)*v[i - 4] - (24*x^2 - 16*x^3)*deriv(v[i - 4], x) + (24 - 8*x + 72*x^2)*v[i - 3] + (8*x^2 + 32*x^3)*deriv(v[i - 3], x) - (22 + 24*x - 36*x^2)*v[i - 2] + (24*x^2 + 16*x^3)*deriv(v[i - 2], x) + (8 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], x)); 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.