Real-rooted atlas

A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)

nonnegative coefficients only rational roots

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 9\,x + 33\,x^{2} + 63\,x^{3} + 66\,x^{4} + 36\,x^{5} + 8\,x^{6}\right)\,P_{n-3}(x) + \left(-3 - 21\,x - 57\,x^{2} - 75\,x^{3} - 48\,x^{4} - 12\,x^{5}\right)\,P_{n-2}(x) + \left(3 + 15\,x + 27\,x^{2} + 21\,x^{3} + 6\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1 + 4\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
00
10
21449
3318362481
464814419296486
5101004008008003202430
61518090024003600288096010935
7212941764588011760141129408268845927
828448313612544313605017650176286727168183708
936648518424192725761451521935361658888294418432708588
1045900810043200151200362880604800691200518400230400460802657205
11551210121007260029040081312016262402323200232320015488006195201126409743085
1266158417424116160522720167270439029766690816836352074342404460544162201627033635075106
13782028243361784648923203212352856627217132544256988162855424022843392124600324153344638976124357194
14912548331242649921457456582982417489472399759366995788893277184932771846783795233918976104366081490944435250179

Computed in your browser ?

Technical details
Node id
n-0ea1c8d30da3abae · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-11aec07ce60b6d61

Observed on the stored rows ?

  • Degrees: -1, -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:  0;
  1:  0;
  2:  1,   4,    4;
  3:  3,  18,   36,    24;
  4:  6,  48,  144,   192,    96;
  5: 10, 100,  400,   800,   800,    320;
  6: 15, 180,  900,  2400,  3600,   2880,    960;
  7: 21, 294, 1764,  5880, 11760,  14112,   9408,   2688;
  8: 28, 448, 3136, 12544, 31360,  50176,  50176,  28672,  7168;
  9: 36, 648, 5184, 24192, 72576, 145152, 193536, 165888, 82944, 18432;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+9*x+33*x^2+63*x^3+66*x^4+36*x^5+8*x^6)*P(n-3,x) - (3+21*x+57*x^2+75*x^3+48*x^4+12*x^5)*P(n-2,x) + (3+15*x+27*x^2+21*x^3+6*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1 + 4*x + 4*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1 + 4*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 9*x + 33*x^2 + 63*x^3 + 66*x^4 + 36*x^5 + 8*x^6)*P[n - 3, x] - (3 + 21*x + 57*x^2 + 75*x^3 + 48*x^4 + 12*x^5)*P[n - 2, x] + (3 + 15*x + 27*x^2 + 21*x^3 + 6*x^4)*P[n - 1, x])/(1 + 3*x + 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] = 0; v[2] = 0; v[3] = 1 + 4*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((1 + 9*x + 33*x^2 + 63*x^3 + 66*x^4 + 36*x^5 + 8*x^6)*v[i - 3] - (3 + 21*x + 57*x^2 + 75*x^3 + 48*x^4 + 12*x^5)*v[i - 2] + (3 + 15*x + 27*x^2 + 21*x^3 + 6*x^4)*v[i - 1])/(1 + 3*x + 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 [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-018032959df2ae1c
  • 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-018032959df2ae1c
  • 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-018032959df2ae1c

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 (67)

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.