Real-rooted atlas

A134991: Triangle of Ward numbers T(n,k) read by rows.

A134991 nonnegative coefficients

Proved for all n: all roots real, in (−1, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A134991
\(P_{n}(x) = \left(1 + x + n x\right)\,P_{n-1}(x) + \left(x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2134
31101526
4125105105236
515649012609452752
6111919189450173251039539208
71246682556980190575270270135135660032
8150122935302995163663540990954729725202702512818912

Show all 15 stored rows

Generated b-files: A134991 ?

Computed in your browser ?

Technical details
Node id
n-3fea9ad1525b48eb · revision 1 · JSON
Definition source
oeis_catalog:A134991
Fingerprint
f6-014ae34d4ff839c2

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,    3;
   3: 1,   10,     15;
   4: 1,   25,    105,     105;
   5: 1,   56,    490,    1260,      945;
   6: 1,  119,   1918,    9450,    17325,     10395;
   7: 1,  246,   6825,   56980,   190575,    270270,     135135;
   8: 1,  501,  22935,  302995,  1636635,   4099095,    4729725,    2027025;
   9: 1, 1012,  74316, 1487200, 12122110,  47507460,   94594500,   91891800,   34459425;
  10: 1, 2035, 235092, 6914908, 81431350, 466876410, 1422280860, 2343240900, 1964187225, 654729075;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) + (x+x^2)*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[(1 + (n + 1)*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
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] = (1 + (n + 1)*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); 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.

A134991: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,    3;
   3: 1,   10,     15;
   4: 1,   25,    105,     105;
   5: 1,   56,    490,    1260,      945;
   6: 1,  119,   1918,    9450,    17325,     10395;
   7: 1,  246,   6825,   56980,   190575,    270270,     135135;
   8: 1,  501,  22935,  302995,  1636635,   4099095,    4729725,    2027025;
   9: 1, 1012,  74316, 1487200, 12122110,  47507460,   94594500,   91891800,   34459425;
  10: 1, 2035, 235092, 6914908, 81431350, 466876410, 1422280860, 2343240900, 1964187225, 654729075;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) + (x+x^2)*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[(1 + (n + 1)*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {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] = (1 + (n + 1)*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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 (-1, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-c13d1846cdae7550
  • 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-c13d1846cdae7550
  • 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-c13d1846cdae7550
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-c13d1846cdae7550

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

Used in 2 proof steps

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

References

  1. [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