Real-rooted atlas

type_b_eulerian → γ (scale 1/4) → D+1

nonnegative coefficients

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

Recurrence
\(\left(-4 + 4\,x + n\right)\,P_{n}(x) = \left(9 - 42\,x + 82\,x^{2} - 68\,x^{3} + 16\,x^{4} - 3\,n + 16\,n x - 46\,n x^{2} + 42\,n x^{3} - 8\,n x^{4} - 2\,n^{2} x + 11\,n^{2} x^{2} - 6\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 38\,x^{2} + 76\,x^{3} - 60\,x^{4} + 16\,x^{5} - 2\,n x + 16\,n x^{2} - 26\,n x^{3} + 12\,n x^{4} - 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-15 + 48\,x - 42\,x^{2} + 8\,x^{3} + 4\,n - 16\,n x + 10\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x + 16\,x^{2} - 8\,x^{3} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2213
36511
41928552
55918061300
61801137662612040
754467657266138515960
816373817675354245681385141120
94917207248719994408360505211391040
101475810952416389528624383013261225052115120000
114428256828135348862486732590309745262702765179625600
121328552912257242838495111043443206521578299832910827027652315174400
1339857514800424433189100951309025176012350230605306578454019936098132172940800
14119573674799271325081825854146600754275212473612300847797841599596075582199360981479480601600

Computed in your browser ?

Technical details
Node id
n-0922e9490abc04e5 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0e92485c0fb0285b

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): 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,      5;
  4:   19,     28,      5;
  5:   59,    180,     61;
  6:  180,   1137,    662,     61;
  7:  544,   6765,   7266,   1385;
  8: 1637,  38176,  75354,  24568,  1385;
  9: 4917, 207248, 719994, 408360, 50521;
%F Formula
The row polynomials satisfy (n-4+4*x)*P(n,x) = -(3*n-9+(2*n^2-16*n+42)*x+(n^3-11*n^2+46*n-82)*x^2+(6*n^2-42*n+68)*x^3+(8*n-16)*x^4)*P(n-2,x) - ((2*n-6)*x+(2*n^2-16*n+38)*x^2-(2*n^2-26*n+76)*x^3-(12*n-60)*x^4-16*x^5)*d/dx P(n-2,x) + (4*n-15+(2*n^2-16*n+48)*x+(10*n-42)*x^2+8*x^3)*P(n-1,x) + ((2*n-8)*x-(2*n-16)*x^2-8*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 - 16*n + 42)*x + (n^3 - 11*n^2 + 46*n - 82)*x^2 + (6*n^2 - 42*n + 68)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] - ((2*n - 6)*x + (2*n^2 - 16*n + 38)*x^2 - (2*n^2 - 26*n + 76)*x^3 - (12*n - 60)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (4*n - 15 + (2*n^2 - 16*n + 48)*x + (10*n - 42)*x^2 + 8*x^3)*P[n - 1, x] + ((2*n - 8)*x - (2*n - 16)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(n - 4 + 4*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 - 16*n + 42)*x + (n^3 - 11*n^2 + 46*n - 82)*x^2 + (6*n^2 - 42*n + 68)*x^3 + (8*n - 16)*x^4)*v[i - 2] - ((2*n - 6)*x + (2*n^2 - 16*n + 38)*x^2 - (2*n^2 - 26*n + 76)*x^3 - (12*n - 60)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (4*n - 15 + (2*n^2 - 16*n + 48)*x + (10*n - 42)*x^2 + 8*x^3)*v[i - 1] + ((2*n - 8)*x - (2*n - 16)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(n - 4 + 4*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-0b45882766291d5a
  • 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-0b45882766291d5a
  • 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-0b45882766291d5a

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.