Real-rooted atlas

A007318 → ×(−(1/2)n+(1/2)n²) → D

palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(-6 + 11\,n - 6\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3} - 3\,n^{2} - 9\,n^{2} x - 9\,n^{2} x^{2} - 3\,n^{2} x^{3} + n^{3} + 3\,n^{3} x + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-9\,n - 18\,n x - 9\,n x^{2} + 12\,n^{2} + 24\,n^{2} x + 12\,n^{2} x^{2} - 3\,n^{3} - 6\,n^{3} x - 3\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(18\,n + 18\,n x - 15\,n^{2} - 15\,n^{2} x + 3\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2 + 2\,x\)
\(P_{3}(x) = 9 + 18\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
2224
3918936
424727224192
55020030020050800
690450900900450902880
71478822205294022058821479408

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-7d6e8eec34fa58e8 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-77a7ed3b1d5f257e

Observed on the stored rows ?

  • Degrees: -1, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   0;
  2:   2,    2;
  3:   9,   18,    9;
  4:  24,   72,   72,    24;
  5:  50,  200,  300,   200,    50;
  6:  90,  450,  900,   900,   450,    90;
  7: 147,  882, 2205,  2940,  2205,   882,  147;
  8: 224, 1568, 4704,  7840,  7840,  4704, 1568,  224;
  9: 324, 2592, 9072, 18144, 22680, 18144, 9072, 2592, 324;
%F Formula
The row polynomials satisfy (n^3-6*n^2+11*n-6)*P(n,x) = (n^3-3*n^2+2*n+(3*n^3-9*n^2+6*n)*x+(3*n^3-9*n^2+6*n)*x^2+(n^3-3*n^2+2*n)*x^3)*P(n-3,x) - (3*n^3-12*n^2+9*n+(6*n^3-24*n^2+18*n)*x+(3*n^3-12*n^2+9*n)*x^2)*P(n-2,x) + (3*n^3-15*n^2+18*n+(3*n^3-15*n^2+18*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2 + 2*x, P(3,x) = 9 + 18*x + 9*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2 + 2*x;
P[3, x_] := 9 + 18*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n^2 + 2*n + (3*n^3 - 9*n^2 + 6*n)*x + (3*n^3 - 9*n^2 + 6*n)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)*P[n - 3, x] - (3*n^3 - 12*n^2 + 9*n + (6*n^3 - 24*n^2 + 18*n)*x + (3*n^3 - 12*n^2 + 9*n)*x^2)*P[n - 2, x] + (3*n^3 - 15*n^2 + 18*n + (3*n^3 - 15*n^2 + 18*n)*x)*P[n - 1, x])/(n^3 - 6*n^2 + 11*n - 6)]];
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 + 2*x; v[4] = 9 + 18*x + 9*x^2; for(n = 4, N, my(i = n + 1); v[i] = ((n^3 - 3*n^2 + 2*n + (3*n^3 - 9*n^2 + 6*n)*x + (3*n^3 - 9*n^2 + 6*n)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)*v[i - 3] - (3*n^3 - 12*n^2 + 9*n + (6*n^3 - 24*n^2 + 18*n)*x + (3*n^3 - 12*n^2 + 9*n)*x^2)*v[i - 2] + (3*n^3 - 15*n^2 + 18*n + (3*n^3 - 15*n^2 + 18*n)*x)*v[i - 1])/(n^3 - 6*n^2 + 11*n - 6)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-b26125b706cf9f72
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-b26125b706cf9f72
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-b26125b706cf9f72

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (59)

Equal up to content, sign, reversal and a small index shift.

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