Real-rooted atlas

Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

A256890 (tail) palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A256890
\(P_{n}(x) = \left(2 + 2\,x + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 4 + 12\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1412420
2852528120
31619641619616840
43268426442644684326720
564227614680264401468022766460480
6128734074652220280220280746527340128604800
725623172357328162396426433601623964357328231722566652800
85127207616378601097844427227908272279081097844416378607207651279833600

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-9c8a3637306e75dc · revision 1 · JSON
Definition source
oeis_catalog:A256890
Fingerprint
f6-9f31c05ba8767dc5

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): 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:    4,     12,        4;
   2:    8,     52,       52,         8;
   3:   16,    196,      416,       196,         16;
   4:   32,    684,     2644,      2644,        684,         32;
   5:   64,   2276,    14680,     26440,      14680,       2276,         64;
   6:  128,   7340,    74652,    220280,     220280,      74652,       7340,        128;
   7:  256,  23172,   357328,   1623964,    2643360,    1623964,     357328,      23172,       256;
   8:  512,  72076,  1637860,  10978444,   27227908,   27227908,   10978444,    1637860,     72076,      512;
   9: 1024, 221860,  7272200,  69632960,  251195000,  381190712,  251195000,   69632960,   7272200,   221860,   1024;
  10: 2048, 677868, 31529260, 420886800, 2133866640, 4677894984, 4677894984, 2133866640, 420886800, 31529260, 677868, 2048;
%F Formula
The row polynomials satisfy P(n,x) = (2+(n+2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 12*x + 4*x^2.
%t Mathematica
P[1, x_] := 4 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (n + 2)*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] = 4 + 12*x + 4*x^2; for(n = 2, N, my(i = n); v[i] = (2 + (n + 2)*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.

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-072c1088d2b33d0a
  • 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-072c1088d2b33d0a
  • 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-072c1088d2b33d0a
  • 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-072c1088d2b33d0a

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 3 proof steps

Same fingerprint (2)

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