Real-rooted atlas

A010054 → reverse → D+1

nonnegative coefficients only rational roots

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

Recurrence
\(\left(-6 + 11\,x - 6\,x^{2} + x^{3} + 11\,n - 12\,n x + 3\,n x^{2} - 6\,n^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(2\,x^{4} - 3\,x^{5} + x^{6} + 2\,n x^{3} - 6\,n x^{4} + 3\,n x^{5} - 3\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-9\,x^{3} + 12\,x^{4} - 3\,x^{5} - 9\,n x^{2} + 24\,n x^{3} - 9\,n x^{4} + 12\,n^{2} x^{2} - 9\,n^{2} x^{3} - 3\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(18\,x^{2} - 15\,x^{3} + 3\,x^{4} + 18\,n x - 30\,n x^{2} + 9\,n x^{3} - 15\,n^{2} x + 9\,n^{2} x^{2} + 3\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)
\(P_{2}(x) = 12\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2012618
300301040
4000601575
5000010521126
60000016828196
700000025236288

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-2ec4ee15f07899eb · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2327141074f67bdf

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): 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: 3,  3;
  2: 0, 12,  6;
  3: 0,  0, 30, 10;
  4: 0,  0,  0, 60,  15;
  5: 0,  0,  0,  0, 105,  21;
  6: 0,  0,  0,  0,   0, 168,  28;
  7: 0,  0,  0,  0,   0,   0, 252,  36;
  8: 0,  0,  0,  0,   0,   0,   0, 360,  45;
  9: 0,  0,  0,  0,   0,   0,   0,   0, 495, 55;
%F Formula
The row polynomials satisfy (n^3-6*n^2+11*n-6+(3*n^2-12*n+11)*x+(3*n-6)*x^2+x^3)*P(n,x) = ((n^3-3*n^2+2*n)*x^3+(3*n^2-6*n+2)*x^4+(3*n-3)*x^5+x^6)*P(n-3,x) - ((3*n^3-12*n^2+9*n)*x^2+(9*n^2-24*n+9)*x^3+(9*n-12)*x^4+3*x^5)*P(n-2,x) + ((3*n^3-15*n^2+18*n)*x+(9*n^2-30*n+18)*x^2+(9*n-15)*x^3+3*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 3*x, P(2,x) = 12*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[2, x_] := 12*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n^3 - 3*n^2 + 2*n)*x^3 + (3*n^2 - 6*n + 2)*x^4 + (3*n - 3)*x^5 + x^6)*P[n - 3, x] - ((3*n^3 - 12*n^2 + 9*n)*x^2 + (9*n^2 - 24*n + 9)*x^3 + (9*n - 12)*x^4 + 3*x^5)*P[n - 2, x] + ((3*n^3 - 15*n^2 + 18*n)*x + (9*n^2 - 30*n + 18)*x^2 + (9*n - 15)*x^3 + 3*x^4)*P[n - 1, x])/(n^3 - 6*n^2 + 11*n - 6 + (3*n^2 - 12*n + 11)*x + (3*n - 6)*x^2 + x^3)]];
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] = 3 + 3*x; v[3] = 12*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (((n^3 - 3*n^2 + 2*n)*x^3 + (3*n^2 - 6*n + 2)*x^4 + (3*n - 3)*x^5 + x^6)*v[i - 3] - ((3*n^3 - 12*n^2 + 9*n)*x^2 + (9*n^2 - 24*n + 9)*x^3 + (9*n - 12)*x^4 + 3*x^5)*v[i - 2] + ((3*n^3 - 15*n^2 + 18*n)*x + (9*n^2 - 30*n + 18)*x^2 + (9*n - 15)*x^3 + 3*x^4)*v[i - 1])/(n^3 - 6*n^2 + 11*n - 6 + (3*n^2 - 12*n + 11)*x + (3*n - 6)*x^2 + x^3)); 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-28592ff7e93ff570
  • 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-28592ff7e93ff570
  • 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-28592ff7e93ff570

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 (10)

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.