Real-rooted atlas

A145596 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence
\(\left(25 + 5\,x^{2} + 65\,n + 11\,n x^{2} + 32\,n^{2} + 7\,n^{2} x^{2} + 4\,n^{3} + n^{3} x^{2}\right)\,P_{n}(x) = \left(75 + 50\,x + 15\,x^{2} + 10\,x^{3} + 95\,n + 102\,n x + 26\,n x^{2} + 21\,n x^{3} + 36\,n^{2} + 64\,n^{2} x + 12\,n^{2} x^{2} + 14\,n^{2} x^{3} + 4\,n^{3} + 12\,n^{3} x + n^{3} x^{2} + 3\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(34\,x - 22\,x^{2} - 8\,x^{3} - 4\,x^{4} + 36\,n x - 30\,n x^{2} - 6\,n x^{4} + 8\,n^{2} x - 8\,n^{2} x^{2} + 2\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0325
1510318
272823462
396095445213
411110285250756737
51318270099456011872574
615280149831362842112017589074
717408289884001117270562058248932266
819570519019908365403376815750354033910115634
9217708745429001039501322649009032340577545011417316
102310121402585668264990444312415998217800620739020583121515516
112513002159316073261818913213201628055117031248605711268413585740135534605
1227163832123286286134033935581265582577527584230060031015014195195198389231420313945
132920304641048794227327308822814171771602065349015459444715715020040023248702821011341574897430
14312480653808008005287100204043844827823071980480685041504164160015979964377104052234039200137516277278570

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Technical details
Node id
n-449f6ebb0d4b8f65 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-525cf34517a99f35

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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): no

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:  3,   2;
  1:  5,  10,    3;
  2:  7,  28,   23,     4;
  3:  9,  60,   95,    44,      5;
  4: 11, 110,  285,   250,     75,      6;
  5: 13, 182,  700,   994,    560,    118,     7;
  6: 15, 280, 1498,  3136,   2842,   1120,   175,     8;
  7: 17, 408, 2898,  8400,  11172,   7056,  2058,   248,    9;
  8: 19, 570, 5190, 19908,  36540,  33768, 15750,  3540,  339,  10;
  9: 21, 770, 8745, 42900, 103950, 132264, 90090, 32340, 5775, 450, 11;
%F Formula
The row polynomials satisfy (4*n^3+32*n^2+65*n+25+(n^3+7*n^2+11*n+5)*x^2)*P(n,x) = (4*n^3+36*n^2+95*n+75+(12*n^3+64*n^2+102*n+50)*x+(n^3+12*n^2+26*n+15)*x^2+(3*n^3+14*n^2+21*n+10)*x^3)*P(n-1,x) + ((8*n^2+36*n+34)*x-(8*n^2+30*n+22)*x^2+(2*n^2-8)*x^3-(2*n^2+6*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 36*n^2 + 95*n + 75 + (12*n^3 + 64*n^2 + 102*n + 50)*x + (n^3 + 12*n^2 + 26*n + 15)*x^2 + (3*n^3 + 14*n^2 + 21*n + 10)*x^3)*P[n - 1, x] + ((8*n^2 + 36*n + 34)*x - (8*n^2 + 30*n + 22)*x^2 + (2*n^2 - 8)*x^3 - (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 32*n^2 + 65*n + 25 + (n^3 + 7*n^2 + 11*n + 5)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 + 36*n^2 + 95*n + 75 + (12*n^3 + 64*n^2 + 102*n + 50)*x + (n^3 + 12*n^2 + 26*n + 15)*x^2 + (3*n^3 + 14*n^2 + 21*n + 10)*x^3)*v[i - 1] + ((8*n^2 + 36*n + 34)*x - (8*n^2 + 30*n + 22)*x^2 + (2*n^2 - 8)*x^3 - (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 32*n^2 + 65*n + 25 + (n^3 + 7*n^2 + 11*n + 5)*x^2)); 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-455a924384122a0c

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

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
  3. [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
  4. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023