Real-rooted atlas

A062145 → D+1

nonnegative coefficients

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

Recurrence
\(\left(2 + 6\,x - 2\,x^{2} + n - 7\,n x + n x^{2} - 4\,n^{2} - n^{2} x + n^{2} x^{2} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(2 + 2\,x - 6\,x^{2} + 2\,x^{3} - 3\,n + n x + 3\,n x^{2} + n x^{3} - 2\,n^{2} - 8\,n^{2} x - 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + 2\,n^{3} + 2\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-x^{2} + 2\,x^{3} - x^{4} - 4\,n x + 4\,n x^{2} + 2\,n x^{3} - 2\,n x^{4} + 2\,n^{2} x + 2\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2549
311301051
41910810520252
529280546280351170
64160019601960630565247
755113455809240567012608423023
871196013545336003381014112231012099528
989316829260101640152460103488314163960165425646
101094860578162679605613305654882772006415264352201805570
111317150106470634920177177024864841801800669240122265100102867610526
12155101641851851381380495495092492409315306509652014864852202201501536431914984
131811404030723028028001256755530126096403002603057912013095225308308037837821840455133276260
1420918928489840536536029409380880719841511590081521405609018009031082480603803262462430940560554611995
15239249907549369777040643242602353791445043838806507689765110205102433431006902095211262160996268428406802301099975

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Technical details
Node id
n-efa728d8a00b7c8f · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-41d66f02e7ef94b4

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   5,    4;
   3:  11,   30,    10;
   4:  19,  108,   105,     20;
   5:  29,  280,   546,    280,     35;
   6:  41,  600,  1960,   1960,    630,     56;
   7:  55, 1134,  5580,   9240,   5670,   1260,     84;
   8:  71, 1960, 13545,  33600,  33810,  14112,   2310,   120;
   9:  89, 3168, 29260, 101640, 152460, 103488,  31416,  3960,  165;
  10: 109, 4860, 57816, 267960, 561330, 565488, 277200, 64152, 6435, 220;
%F Formula
The row polynomials satisfy (n^4-4*n^2+n+2+(2*n^3-n^2-7*n+6)*x+(n^2+n-2)*x^2)*P(n,x) = (n^4+2*n^3-2*n^2-3*n+2+(3*n^4+2*n^3-8*n^2+n+2)*x+(6*n^3-3*n^2+3*n-6)*x^2+(3*n^2+n+2)*x^3)*P(n-1,x) + ((2*n^3+2*n^2-4*n)*x-(2*n^3-2*n^2-4*n+1)*x^2-(4*n^2-2*n-2)*x^3-(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 2*n^3 - 2*n^2 - 3*n + 2 + (3*n^4 + 2*n^3 - 8*n^2 + n + 2)*x + (6*n^3 - 3*n^2 + 3*n - 6)*x^2 + (3*n^2 + n + 2)*x^3)*P[n - 1, x] + ((2*n^3 + 2*n^2 - 4*n)*x - (2*n^3 - 2*n^2 - 4*n + 1)*x^2 - (4*n^2 - 2*n - 2)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^2 + n + 2 + (2*n^3 - n^2 - 7*n + 6)*x + (n^2 + n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^4 + 2*n^3 - 2*n^2 - 3*n + 2 + (3*n^4 + 2*n^3 - 8*n^2 + n + 2)*x + (6*n^3 - 3*n^2 + 3*n - 6)*x^2 + (3*n^2 + n + 2)*x^3)*v[i - 1] + ((2*n^3 + 2*n^2 - 4*n)*x - (2*n^3 - 2*n^2 - 4*n + 1)*x^2 - (4*n^2 - 2*n - 2)*x^3 - (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^2 + n + 2 + (2*n^3 - n^2 - 7*n + 6)*x + (n^2 + n - 2)*x^2)); 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-df4a805affbb39de
  • 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-df4a805affbb39de
  • 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-df4a805affbb39de
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-df4a805affbb39de

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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.