Real-rooted atlas

A108838 → ⊙ C(n+2,k+1) (shift +2)

A197208 (tail) palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A197208
\(\left(1 + \frac{6}{5}\,n + \frac{1}{5}\,n^{2}\right)\,P_{n}(x) = \left(4 + 4\,x + \frac{7}{5}\,n + \frac{28}{5}\,n x - \frac{1}{5}\,n^{2} + \frac{8}{5}\,n^{2} x\right)\,P_{n-1}(x) + \left(\frac{21}{5}\,x - \frac{21}{5}\,x^{2} + \frac{9}{5}\,n x - \frac{9}{5}\,n x^{2}\right)\,P_{n-1}'(x) + \left(\frac{6}{5} + \frac{12}{5}\,x + \frac{6}{5}\,x^{2} + \frac{8}{5}\,n + \frac{16}{5}\,n x + \frac{8}{5}\,n x^{2} + \frac{2}{5}\,n^{2} + \frac{4}{5}\,n^{2} x + \frac{2}{5}\,n^{2} x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 30 + 120\,x + 30\,x^{2}\)
\(P_{2}(x) = 60 + 600\,x + 600\,x^{2} + 60\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
13012030180
260600600601320
31052100525021001059660
416858802940029400588016870896
52521411212348024696012348014112252522648
63603024042336014817601481760423360302403603871440
7495594001247400698544012224520698544012474005940049528809990
86601089003267000274428007683984076839840274428003267000108900660215318400

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-05b91207a1ba3acf · revision 1 · JSON
Definition source
oeis_catalog:A197208
Fingerprint
f6-509399d8a74ba170

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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): 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   30,    120,       30;
   2:   60,    600,      600,        60;
   3:  105,   2100,     5250,      2100,        105;
   4:  168,   5880,    29400,     29400,       5880,        168;
   5:  252,  14112,   123480,    246960,     123480,      14112,        252;
   6:  360,  30240,   423360,   1481760,    1481760,     423360,      30240,        360;
   7:  495,  59400,  1247400,   6985440,   12224520,    6985440,    1247400,      59400,       495;
   8:  660, 108900,  3267000,  27442800,   76839840,   76839840,   27442800,    3267000,    108900,      660;
   9:  858, 188760,  7786350,  93436200,  392432040,  627891264,  392432040,   93436200,   7786350,   188760,    858;
  10: 1092, 312312, 17177160, 283423140, 1700538840, 4081293216, 4081293216, 1700538840, 283423140, 17177160, 312312, 1092;
%F Formula
The row polynomials satisfy ((1/5)*n^2+(6/5)*n+1)*P(n,x) = -((1/5)*n^2-(7/5)*n-4-((8/5)*n^2+(28/5)*n+4)*x)*P(n-1,x) + (((9/5)*n+21/5)*x-((9/5)*n+21/5)*x^2)*d/dx P(n-1,x) + ((2/5)*n^2+(8/5)*n+6/5+((4/5)*n^2+(16/5)*n+12/5)*x+((2/5)*n^2+(8/5)*n+6/5)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 30 + 120*x + 30*x^2, P(2,x) = 60 + 600*x + 600*x^2 + 60*x^3.
%t Mathematica
P[1, x_] := 30 + 120*x + 30*x^2;
P[2, x_] := 60 + 600*x + 600*x^2 + 60*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((1/5)*n^2 - (7/5)*n - 4 - ((8/5)*n^2 + (28/5)*n + 4)*x)*P[n - 1, x] + (((9/5)*n + 21/5)*x - ((9/5)*n + 21/5)*x^2)*D[P[n - 1, x], x] + ((2/5)*n^2 + (8/5)*n + 6/5 + ((4/5)*n^2 + (16/5)*n + 12/5)*x + ((2/5)*n^2 + (8/5)*n + 6/5)*x^2)*P[n - 2, x])/((1/5)*n^2 + (6/5)*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 30 + 120*x + 30*x^2; v[2] = 60 + 600*x + 600*x^2 + 60*x^3; for(n = 3, N, my(i = n); v[i] = (-((1/5)*n^2 - (7/5)*n - 4 - ((8/5)*n^2 + (28/5)*n + 4)*x)*v[i - 1] + (((9/5)*n + 21/5)*x - ((9/5)*n + 21/5)*x^2)*deriv(v[i - 1], x) + ((2/5)*n^2 + (8/5)*n + 6/5 + ((4/5)*n^2 + (16/5)*n + 12/5)*x + ((2/5)*n^2 + (8/5)*n + 6/5)*x^2)*v[i - 2])/((1/5)*n^2 + (6/5)*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-aaeb3397827b8bbd
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-aaeb3397827b8bbd
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-aaeb3397827b8bbd

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 (1)

Equal up to content, sign, reversal and a small index shift.

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023