Real-rooted atlas

A102365 → 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
\(\left(-4 + 2\,x + n\right)\,P_{n}(x) = \left(9 - 30\,x + 49\,x^{2} - 28\,x^{3} + 4\,x^{4} - 3\,n + 14\,n x - 36\,n x^{2} + 19\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 10\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 31\,x^{2} + 36\,x^{3} - 15\,x^{4} + 2\,x^{5} - 2\,n x + 15\,n x^{2} - 13\,n x^{3} + 3\,n x^{4} - 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-15 + 32\,x - 17\,x^{2} + 2\,x^{3} + 4\,n - 14\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x + 8\,x^{2} - 2\,x^{3} + 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2213
367114
4194818186
559316240411657
6180193328159788815995
75441110329370187703570183163541
8163760952277488307750106295122043741766701
9491732440024283964434678259542554382440020757110372418
1014758169083320073198576670925423812819041351260171112768615241155456282
11442828690287159082035692412852100028905255639661712706142411884110399894305912556263613
121328554424850412220921307818213015167026890151406716290050501109967928145605252725512364806130145751233841
133985752238996129173725404841848841752578025970903164946298201705226268484194964389647103927752266727803789240122731885293272489
1411957361128266701676851268058736782284943738550579150648215586959450449643876641852934945640326199171276269811886259609543191153967424560118415251478239

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Technical details
Node id
n-f8fcbabbad3b1ee6 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-60a010a75360b171

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    1;
  1:    1;
  2:    2,      1;
  3:    6,      7,       1;
  4:   19,     48,      18,       1;
  5:   59,    316,     240,      41,       1;
  6:  180,   1933,    2815,     978,      88,      1;
  7:  544,  11103,   29370,   18770,    3570,    183,     1;
  8: 1637,  60952,  277488,  307750,  106295,  12204,   374,   1;
  9: 4917, 324400, 2428396, 4434678, 2595425, 543824, 40020, 757, 1;
%F Formula
The row polynomials satisfy (n-4+2*x)*P(n,x) = -(3*n-9+(2*n^2-14*n+30)*x+(n^3-10*n^2+36*n-49)*x^2+(3*n^2-19*n+28)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((2*n-6)*x+(2*n^2-15*n+31)*x^2-(n^2-13*n+36)*x^3-(3*n-15)*x^4-2*x^5)*d/dx P(n-2,x) + (4*n-15+(2*n^2-14*n+32)*x+(5*n-17)*x^2+2*x^3)*P(n-1,x) + ((2*n-8)*x-(n-8)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 9 + (2*n^2 - 14*n + 30)*x + (n^3 - 10*n^2 + 36*n - 49)*x^2 + (3*n^2 - 19*n + 28)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((2*n - 6)*x + (2*n^2 - 15*n + 31)*x^2 - (n^2 - 13*n + 36)*x^3 - (3*n - 15)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (4*n - 15 + (2*n^2 - 14*n + 32)*x + (5*n - 17)*x^2 + 2*x^3)*P[n - 1, x] + ((2*n - 8)*x - (n - 8)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 4 + 2*x)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 9 + (2*n^2 - 14*n + 30)*x + (n^3 - 10*n^2 + 36*n - 49)*x^2 + (3*n^2 - 19*n + 28)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((2*n - 6)*x + (2*n^2 - 15*n + 31)*x^2 - (n^2 - 13*n + 36)*x^3 - (3*n - 15)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (4*n - 15 + (2*n^2 - 14*n + 32)*x + (5*n - 17)*x^2 + 2*x^3)*v[i - 1] + ((2*n - 8)*x - (n - 8)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 4 + 2*x)); 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-ff253c56ca542933
  • 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-ff253c56ca542933
  • 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-ff253c56ca542933
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-ff253c56ca542933

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.