Real-rooted atlas

A094305 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(27 + 37\,x + 12\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(37\,x + 24\,x^{2} + 3\,x^{3}\right)\,P_{n-3}'(x) + \left(12\,x^{2} + 3\,x^{3}\right)\,P_{n-3}''(x) + \left(x^{3}\right)\,P_{n-3}'''(x) + \left(-27 - 21\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(-21\,x - 6\,x^{2}\right)\,P_{n-2}'(x) + \left(-3\,x^{2}\right)\,P_{n-2}''(x) + \left(9 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 3\,x\)
\(P_{2}(x) = 54 + 42\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19312
254426102
327037012010770
4121526251455270155580
55103164011386043055252140215
62041294276114268529201064092428293468
778732511092855792555156163800231841512362189304

All 15 rows

Technical details
Node id
n-0446d4cb1fe0a9ca · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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:       9,        3;
  2:      54,       42,        6;
  3:     270,      370,      120,       10;
  4:    1215,     2625,     1455,      270,       15;
  5:    5103,    16401,    13860,     4305,      525,      21;
  6:   20412,    94276,   114268,    52920,    10640,     924,      28;
  7:   78732,   511092,   855792,   555156,   163800,   23184,    1512,    36;
  8:  295245,  2653875,  5987565,  5233410,  2127195,  436590,   45990,  2340,   45;
  9: 1082565, 13335355, 39834300, 45696475, 24595725, 6868785, 1039500, 84810, 3465, 55;
%F Formula
The row polynomials satisfy P(n,x) = (27+37*x+12*x^2+x^3)*P(n-3,x) + (37*x+24*x^2+3*x^3)*d/dx P(n-3,x) + (12*x^2+3*x^3)*d^2/dx^2 P(n-3,x) + x^3*d^3/dx^3 P(n-3,x) - (27+21*x+3*x^2)*P(n-2,x) - (21*x+6*x^2)*d/dx P(n-2,x) - 3*x^2*d^2/dx^2 P(n-2,x) + (9+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 9 + 3*x, P(2,x) = 54 + 42*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 3*x;
P[2, x_] := 54 + 42*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 37*x + 12*x^2 + x^3)*P[n - 3, x] + (37*x + 24*x^2 + 3*x^3)*D[P[n - 3, x], x] + (12*x^2 + 3*x^3)*D[P[n - 3, x], {x, 2}] + x^3*D[P[n - 3, x], {x, 3}] - (27 + 21*x + 3*x^2)*P[n - 2, x] - (21*x + 6*x^2)*D[P[n - 2, x], x] - 3*x^2*D[P[n - 2, x], {x, 2}] + (9 + 3*x)*P[n - 1, x] + 3*x*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] = 9 + 3*x; v[3] = 54 + 42*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (27 + 37*x + 12*x^2 + x^3)*v[i - 3] + (37*x + 24*x^2 + 3*x^3)*deriv(v[i - 3], x) + (12*x^2 + 3*x^3)*deriv(deriv(v[i - 3], x), x) + x^3*deriv(deriv(deriv(v[i - 3], x), x), x) - (27 + 21*x + 3*x^2)*v[i - 2] - (21*x + 6*x^2)*deriv(v[i - 2], x) - 3*x^2*deriv(deriv(v[i - 2], x), x) + (9 + 3*x)*v[i - 1] + 3*x*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

  • 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.