Real-rooted atlas

A007318 → ×(1+(3/2)n+(1/2)n²) → 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
\(\left(2\,x - 3\,x^{2} + x^{3} + 2\,n - 6\,n x + 3\,n x^{2} - 3\,n^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-x - 3\,x^{2} - 2\,x^{3} + 2\,x^{4} + 3\,x^{5} + x^{6} - n - 3\,n x + 8\,n x^{3} + 9\,n x^{4} + 3\,n x^{5} + 3\,n^{2} x + 9\,n^{2} x^{2} + 9\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{3} + 3\,n^{3} x + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-3}(x) + \left(6\,x + 15\,x^{2} + 9\,x^{3} - 3\,x^{4} - 3\,x^{5} + 6\,n + 18\,n x + 9\,n x^{2} - 12\,n x^{3} - 9\,n x^{4} + 3\,n^{2} - 3\,n^{2} x - 15\,n^{2} x^{2} - 9\,n^{2} x^{3} - 3\,n^{3} - 6\,n^{3} x - 3\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6 + 3\,x - 9\,x^{2} - 3\,x^{3} + 3\,x^{4} - 3\,n - 15\,n x - 3\,n x^{2} + 9\,n x^{3} - 6\,n^{2} + 3\,n^{2} x + 9\,n^{2} x^{2} + 3\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)
\(P_{2}(x) = 18 + 24\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
21824648
340906010200
47524027012015720
5126525840630210212352
61961008210022401260336287168
7288176445366300504022685043620736

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-7a1376d2a2b6b46f · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e99c3824c4864868

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 ?

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:   1;
  1:   6,    3;
  2:  18,   24,     6;
  3:  40,   90,    60,    10;
  4:  75,  240,   270,   120,    15;
  5: 126,  525,   840,   630,   210,    21;
  6: 196, 1008,  2100,  2240,  1260,   336,    28;
  7: 288, 1764,  4536,  6300,  5040,  2268,   504,   36;
  8: 405, 2880,  8820, 15120, 15750, 10080,  3780,  720,  45;
  9: 550, 4455, 15840, 32340, 41580, 34650, 18480, 5940, 990, 55;
%F Formula
The row polynomials satisfy (n^3-3*n^2+2*n+(3*n^2-6*n+2)*x+(3*n-3)*x^2+x^3)*P(n,x) = (n^3-n+(3*n^3+3*n^2-3*n-1)*x+(3*n^3+9*n^2-3)*x^2+(n^3+9*n^2+8*n-2)*x^3+(3*n^2+9*n+2)*x^4+(3*n+3)*x^5+x^6)*P(n-3,x) - (3*n^3-3*n^2-6*n+(6*n^3+3*n^2-18*n-6)*x+(3*n^3+15*n^2-9*n-15)*x^2+(9*n^2+12*n-9)*x^3+(9*n+3)*x^4+3*x^5)*P(n-2,x) + (3*n^3-6*n^2-3*n+6+(3*n^3+3*n^2-15*n+3)*x+(9*n^2-3*n-9)*x^2+(9*n-3)*x^3+3*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 3*x, P(2,x) = 18 + 24*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[2, x_] := 18 + 24*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n + (3*n^3 + 3*n^2 - 3*n - 1)*x + (3*n^3 + 9*n^2 - 3)*x^2 + (n^3 + 9*n^2 + 8*n - 2)*x^3 + (3*n^2 + 9*n + 2)*x^4 + (3*n + 3)*x^5 + x^6)*P[n - 3, x] - (3*n^3 - 3*n^2 - 6*n + (6*n^3 + 3*n^2 - 18*n - 6)*x + (3*n^3 + 15*n^2 - 9*n - 15)*x^2 + (9*n^2 + 12*n - 9)*x^3 + (9*n + 3)*x^4 + 3*x^5)*P[n - 2, x] + (3*n^3 - 6*n^2 - 3*n + 6 + (3*n^3 + 3*n^2 - 15*n + 3)*x + (9*n^2 - 3*n - 9)*x^2 + (9*n - 3)*x^3 + 3*x^4)*P[n - 1, x])/(n^3 - 3*n^2 + 2*n + (3*n^2 - 6*n + 2)*x + (3*n - 3)*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] = 6 + 3*x; v[3] = 18 + 24*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - n + (3*n^3 + 3*n^2 - 3*n - 1)*x + (3*n^3 + 9*n^2 - 3)*x^2 + (n^3 + 9*n^2 + 8*n - 2)*x^3 + (3*n^2 + 9*n + 2)*x^4 + (3*n + 3)*x^5 + x^6)*v[i - 3] - (3*n^3 - 3*n^2 - 6*n + (6*n^3 + 3*n^2 - 18*n - 6)*x + (3*n^3 + 15*n^2 - 9*n - 15)*x^2 + (9*n^2 + 12*n - 9)*x^3 + (9*n + 3)*x^4 + 3*x^5)*v[i - 2] + (3*n^3 - 6*n^2 - 3*n + 6 + (3*n^3 + 3*n^2 - 15*n + 3)*x + (9*n^2 - 3*n - 9)*x^2 + (9*n - 3)*x^3 + 3*x^4)*v[i - 1])/(n^3 - 3*n^2 + 2*n + (3*n^2 - 6*n + 2)*x + (3*n - 3)*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-1312db2a2710f639
  • 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-1312db2a2710f639
  • 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-1312db2a2710f639

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 (27)

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.