Real-rooted atlas

A063967 → reverse

nonnegative coefficients

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

Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 - 7\,x - 10\,x^{2} + n + 4\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
31326
4157316
5171615544
6192943308120
711146951045813328
81136717927123510921896
911592303591705506201342448
101171214751140174617171051365556688
11119154703201037804759397921236558918272
1212119199533097405114251220188704194116414449920
1312323213595161134272462032165298091916681362052233136384
1412527718037706228924876175615856007004640366155463593377372608
151273262335111003711890241162423218000217620159387832142932762556101017984

Generated b-files: A063967 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-305352172c67c5f0 · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-4d750740fbfda5f4

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1,  1;
   3: 1,  3,   2;
   4: 1,  5,   7,   3;
   5: 1,  7,  16,  15,    5;
   6: 1,  9,  29,  43,   30,    8;
   7: 1, 11,  46,  95,  104,   58,   13;
   8: 1, 13,  67, 179,  271,  235,  109,   21;
   9: 1, 15,  92, 303,  591,  705,  506,  201,  34;
  10: 1, 17, 121, 475, 1140, 1746, 1717, 1051, 365, 55;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n-1+(4*n-7)*x+(5*n-10)*x^2)*P(n-1,x) - (x+6*x^2+5*x^3)*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 - 1 + (4*n - 7)*x + (5*n - 10)*x^2)*P[n - 1, x] - (x + 6*x^2 + 5*x^3)*D[P[n - 1, x], x])/(n - 1)]];
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 - 1 + (4*n - 7)*x + (5*n - 10)*x^2)*v[i - 1] - (x + 6*x^2 + 5*x^3)*deriv(v[i - 1], x))/(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.

A063967: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1:  1,   1;
  2:  2,   3,    1;
  3:  3,   7,    5,    1;
  4:  5,  15,   16,    7,    1;
  5:  8,  30,   43,   29,    9,    1;
  6: 13,  58,  104,   95,   46,   11,   1;
  7: 21, 109,  235,  271,  179,   67,  13,   1;
  8: 34, 201,  506,  705,  591,  303,  92,  15,  1;
  9: 55, 365, 1051, 1717, 1746, 1140, 475, 121, 17, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (n+(4*n-3)*x+(5*n-5)*x^2)*P(n-1,x) - (x+6*x^2+5*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (4*n - 3)*x + (5*n - 5)*x^2)*P[n - 1, x] - (x + 6*x^2 + 5*x^3)*D[P[n - 1, x], x])/(n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + (4*n - 3)*x + (5*n - 5)*x^2)*v[i - 1] - (x + 6*x^2 + 5*x^3)*deriv(v[i - 1], x))/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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 >= 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-2c112d9ec33e1fb9
  • 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-2c112d9ec33e1fb9
  • 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-2c112d9ec33e1fb9

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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741