Real-rooted atlas

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
26241848
310609040200
41512027024075720
5212106308405251262352
62833612602240210010081967168
7365042268504063004536176428820736
84572037801008015750151208820288040557600
955990594018480346504158032340158404455550154880
1066132089103168069300997929702063360267306600726405504
1178171612870514801287002162162522522059201158304290094389361038336
129121841801880080225225432432588588576576405405200200660661310411832609152
13105273024570120120375375810810126126014414401216215750750330330982801774514706451200
1412033603276017472060060014414402522520329472032432402402400132132052416014196023520180015728640

Computed in your browser ?

Technical details
Node id
n-751f2fc33fb80fef · revision 1 · JSON
Definition source
derived:theta_plus_one
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:  3,   6;
  2:  6,  24,   18;
  3: 10,  60,   90,    40;
  4: 15, 120,  270,   240,    75;
  5: 21, 210,  630,   840,   525,   126;
  6: 28, 336, 1260,  2240,  2100,  1008,   196;
  7: 36, 504, 2268,  5040,  6300,  4536,  1764,   288;
  8: 45, 720, 3780, 10080, 15750, 15120,  8820,  2880,  405;
  9: 55, 990, 5940, 18480, 34650, 41580, 32340, 15840, 4455, 550;
%F Formula
The row polynomials satisfy (1+(3*n-3)*x+(3*n^2-6*n+2)*x^2+(n^3-3*n^2+2*n)*x^3)*P(n,x) = (1+(3*n+3)*x+(3*n^2+9*n+2)*x^2+(n^3+9*n^2+8*n-2)*x^3+(3*n^3+9*n^2-3)*x^4+(3*n^3+3*n^2-3*n-1)*x^5+(n^3-n)*x^6)*P(n-3,x) - (3+(9*n+3)*x+(9*n^2+12*n-9)*x^2+(3*n^3+15*n^2-9*n-15)*x^3+(6*n^3+3*n^2-18*n-6)*x^4+(3*n^3-3*n^2-6*n)*x^5)*P(n-2,x) + (3+(9*n-3)*x+(9*n^2-3*n-9)*x^2+(3*n^3+3*n^2-15*n+3)*x^3+(3*n^3-6*n^2-3*n+6)*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 6*x, P(2,x) = 6 + 24*x + 18*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[2, x_] := 6 + 24*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (3*n + 3)*x + (3*n^2 + 9*n + 2)*x^2 + (n^3 + 9*n^2 + 8*n - 2)*x^3 + (3*n^3 + 9*n^2 - 3)*x^4 + (3*n^3 + 3*n^2 - 3*n - 1)*x^5 + (n^3 - n)*x^6)*P[n - 3, x] - (3 + (9*n + 3)*x + (9*n^2 + 12*n - 9)*x^2 + (3*n^3 + 15*n^2 - 9*n - 15)*x^3 + (6*n^3 + 3*n^2 - 18*n - 6)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*P[n - 2, x] + (3 + (9*n - 3)*x + (9*n^2 - 3*n - 9)*x^2 + (3*n^3 + 3*n^2 - 15*n + 3)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*P[n - 1, x])/(1 + (3*n - 3)*x + (3*n^2 - 6*n + 2)*x^2 + (n^3 - 3*n^2 + 2*n)*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 + 6*x; v[3] = 6 + 24*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((1 + (3*n + 3)*x + (3*n^2 + 9*n + 2)*x^2 + (n^3 + 9*n^2 + 8*n - 2)*x^3 + (3*n^3 + 9*n^2 - 3)*x^4 + (3*n^3 + 3*n^2 - 3*n - 1)*x^5 + (n^3 - n)*x^6)*v[i - 3] - (3 + (9*n + 3)*x + (9*n^2 + 12*n - 9)*x^2 + (3*n^3 + 15*n^2 - 9*n - 15)*x^3 + (6*n^3 + 3*n^2 - 18*n - 6)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*v[i - 2] + (3 + (9*n - 3)*x + (9*n^2 - 3*n - 9)*x^2 + (3*n^3 + 3*n^2 - 15*n + 3)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*v[i - 1])/(1 + (3*n - 3)*x + (3*n^2 - 6*n + 2)*x^2 + (n^3 - 3*n^2 + 2*n)*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-df5c3bb3c9c4e6f1
  • 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-df5c3bb3c9c4e6f1
  • 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-df5c3bb3c9c4e6f1

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.