Real-rooted atlas

A054142 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(12\,x^{2} + x^{3}\right)\,P_{n-6}(x) + \left(4\,x^{3}\right)\,P_{n-6}'(x) + \left(-24\,x - 15\,x^{2}\right)\,P_{n-5}(x) + \left(-12\,x^{2} - 4\,x^{3}\right)\,P_{n-5}'(x) + \left(12 + 3\,x - 3\,x^{2}\right)\,P_{n-4}(x) + \left(12\,x\right)\,P_{n-4}'(x) + \left(27 + 10\,x\right)\,P_{n-3}(x) + \left(-4 - 4\,x + 8\,x^{2}\right)\,P_{n-3}'(x) + \left(-11 + 3\,x\right)\,P_{n-2}(x) + \left(8 + 12\,x\right)\,P_{n-2}'(x) + \left(5\right)\,P_{n-1}(x) + \left(-4 - 4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)
\(P_{3}(x) = 1 + 12\,x\)
\(P_{4}(x) = 1 + 30\,x + 3\,x^{2}\)
\(P_{5}(x) = 1 + 56\,x + 45\,x^{2}\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2123
311213
4130334
515645102
61902104305
71132630112875

All 15 rows

Technical details
Node id
n-97d49116a949e316 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e02f16c4209dfad4

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 1;
  2: 1,   2;
  3: 1,  12;
  4: 1,  30,    3;
  5: 1,  56,   45;
  6: 1,  90,  210,    4;
  7: 1, 132,  630,  112;
  8: 1, 182, 1485,  840,   5;
  9: 1, 240, 3003, 3696, 225;
%F Formula
The row polynomials satisfy P(n,x) = (12*x^2+x^3)*P(n-6,x) + 4*x^3*d/dx P(n-6,x) - (24*x+15*x^2)*P(n-5,x) - (12*x^2+4*x^3)*d/dx P(n-5,x) + (12+3*x-3*x^2)*P(n-4,x) + 12*x*d/dx P(n-4,x) + (27+10*x)*P(n-3,x) - (4+4*x-8*x^2)*d/dx P(n-3,x) - (11-3*x)*P(n-2,x) + (8+12*x)*d/dx P(n-2,x) + 5*P(n-1,x) - (4+4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 1 + 12*x, P(4,x) = 1 + 30*x + 3*x^2, P(5,x) = 1 + 56*x + 45*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 1 + 12*x;
P[4, x_] := 1 + 30*x + 3*x^2;
P[5, x_] := 1 + 56*x + 45*x^2;
P[n_, x_] := P[n, x] = Expand[(12*x^2 + x^3)*P[n - 6, x] + 4*x^3*D[P[n - 6, x], x] - (24*x + 15*x^2)*P[n - 5, x] - (12*x^2 + 4*x^3)*D[P[n - 5, x], x] + (12 + 3*x - 3*x^2)*P[n - 4, x] + 12*x*D[P[n - 4, x], x] + (27 + 10*x)*P[n - 3, x] - (4 + 4*x - 8*x^2)*D[P[n - 3, x], x] - (11 - 3*x)*P[n - 2, x] + (8 + 12*x)*D[P[n - 2, x], x] + 5*P[n - 1, x] - (4 + 4*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] = 1; v[3] = 1 + 2*x; v[4] = 1 + 12*x; v[5] = 1 + 30*x + 3*x^2; v[6] = 1 + 56*x + 45*x^2; for(n = 6, N, my(i = n + 1); v[i] = (12*x^2 + x^3)*v[i - 6] + 4*x^3*deriv(v[i - 6], x) - (24*x + 15*x^2)*v[i - 5] - (12*x^2 + 4*x^3)*deriv(v[i - 5], x) + (12 + 3*x - 3*x^2)*v[i - 4] + 12*x*deriv(v[i - 4], x) + (27 + 10*x)*v[i - 3] - (4 + 4*x - 8*x^2)*deriv(v[i - 3], x) - (11 - 3*x)*v[i - 2] + (8 + 12*x)*deriv(v[i - 2], x) + 5*v[i - 1] - (4 + 4*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

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  5. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  8. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023