Real-rooted atlas

A010054 → reverse → (θ+1)²

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
\(\left(4\,n^{2} - 12\,n^{3} + 13\,n^{4} - 6\,n^{5} + n^{6}\right)\,P_{n}(x) = \left(n^{2} x^{3} - 2\,n^{4} x^{3} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(-12\,n^{2} x^{2} - 12\,n^{3} x^{2} + 9\,n^{4} x^{2} + 6\,n^{5} x^{2} - 3\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(12\,x - 12\,n x - 21\,n^{2} x + 24\,n^{3} x + 6\,n^{4} x - 12\,n^{5} x + 3\,n^{6} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12\,x\)
\(P_{2}(x) = 54\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
101212
2005454
3000160160
40000375375
500000756756
600000013721372
7000000023042304

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-e3a53e103103e79e · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-edf6954a4cd8adfc

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: 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 ?

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: 1;
  1: 0, 12;
  2: 0,  0, 54;
  3: 0,  0,  0, 160;
  4: 0,  0,  0,   0, 375;
  5: 0,  0,  0,   0,   0, 756;
  6: 0,  0,  0,   0,   0,   0, 1372;
  7: 0,  0,  0,   0,   0,   0,    0, 2304;
  8: 0,  0,  0,   0,   0,   0,    0,    0, 3645;
  9: 0,  0,  0,   0,   0,   0,    0,    0,    0, 5500;
%F Formula
The row polynomials satisfy (n^6-6*n^5+13*n^4-12*n^3+4*n^2)*P(n,x) = (n^6-2*n^4+n^2)*x^3*P(n-3,x) - (3*n^6-6*n^5-9*n^4+12*n^3+12*n^2)*x^2*P(n-2,x) + (3*n^6-12*n^5+6*n^4+24*n^3-21*n^2-12*n+12)*x*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 12*x, P(2,x) = 54*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 12*x;
P[2, x_] := 54*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^6 - 2*n^4 + n^2)*x^3*P[n - 3, x] - (3*n^6 - 6*n^5 - 9*n^4 + 12*n^3 + 12*n^2)*x^2*P[n - 2, x] + (3*n^6 - 12*n^5 + 6*n^4 + 24*n^3 - 21*n^2 - 12*n + 12)*x*P[n - 1, x])/(n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)]];
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] = 12*x; v[3] = 54*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n^6 - 2*n^4 + n^2)*x^3*v[i - 3] - (3*n^6 - 6*n^5 - 9*n^4 + 12*n^3 + 12*n^2)*x^2*v[i - 2] + (3*n^6 - 12*n^5 + 6*n^4 + 24*n^3 - 21*n^2 - 12*n + 12)*x*v[i - 1])/(n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*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)

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-e78403b3b87fde04
  • 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-e78403b3b87fde04
  • 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-e78403b3b87fde04

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 (75)

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