Real-rooted atlas

A003506 → D → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(4\,n + 5\,n x + n x^{2} - 4\,n^{2} - 5\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(5\,n x + 2\,n x^{2} - 5\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(-8\,n - 4\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,n x + 2\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
222
312618
4486012120
516038018020740
648019501650420304530
7134488621197052508404228308
8358437240756565096013720151256182728
99216148248436968424872171360312482520721224504
1023040567450237015032016601816290487620642603960908534520
1156320210881012280950224625501723260063917701224300122100594011061885450
12135168765930061479132149512440151029780743699881942340427878402178008580132466623564
13319488273154442996805249561397561247634960793732368271531260526074125868720368940120121563655211040
147454729597679014303771305927109188984894110279377217923460312856869164296129843714115815705985981638018229712389070
Technical details
Node id
n-ecf64da7365c1c1d · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:    0;
  2:    2;
  3:   12,      6;
  4:   48,     60,     12;
  5:  160,    380,    180,     20;
  6:  480,   1950,   1650,    420,     30;
  7: 1344,   8862,  11970,   5250,    840,    42;
  8: 3584,  37240,  75656,  50960,  13720,  1512,   56;
  9: 9216, 148248, 436968, 424872, 171360, 31248, 2520, 72;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(4*n^2-4*n+(5*n^2-5*n)*x+(n^2-n)*x^2)*P(n-2,x) - ((5*n^2-5*n)*x+(2*n^2-2*n)*x^2)*d/dx P(n-2,x) - (n^2-n)*x^2*d^2/dx^2 P(n-2,x) + (4*n^2-8*n+(2*n^2-4*n)*x)*P(n-1,x) + (2*n^2-4*n)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 4*n + (5*n^2 - 5*n)*x + (n^2 - n)*x^2)*P[n - 2, x] - ((5*n^2 - 5*n)*x + (2*n^2 - 2*n)*x^2)*D[P[n - 2, x], x] - (n^2 - n)*x^2*D[P[n - 2, x], {x, 2}] + (4*n^2 - 8*n + (2*n^2 - 4*n)*x)*P[n - 1, x] + (2*n^2 - 4*n)*x*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
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] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 4*n + (5*n^2 - 5*n)*x + (n^2 - n)*x^2)*v[i - 2] - ((5*n^2 - 5*n)*x + (2*n^2 - 2*n)*x^2)*deriv(v[i - 2], x) - (n^2 - n)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n^2 - 8*n + (2*n^2 - 4*n)*x)*v[i - 1] + (2*n^2 - 4*n)*x*deriv(v[i - 1], x))/(n^2 - 3*n + 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [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
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.