Real-rooted atlas

A144698: Triangle of 4-Eulerian numbers.

A144698 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 A144698
\(P_{n}(x) = \left(1 + 3\,x + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
21131630
313211364210
41715318212561680
51150209064705385102415120
6130974704051065745330694096151200
7162825191221800612295592884194017163841663200

Show all 15 stored rows

Generated b-files: A001720 (row sums), A144698 ?

Computed in your browser ?

Technical details
Node id
n-7e652d8a57b6dcff · revision 1 · JSON
Definition source
oeis_catalog:A144698
Fingerprint
f6-197c535f804b8fcf

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 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 ?

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,    4;
  2: 1,   13,     16;
  3: 1,   32,    113,      64;
  4: 1,   71,    531,     821,      256;
  5: 1,  150,   2090,    6470,     5385,     1024;
  6: 1,  309,   7470,   40510,    65745,    33069,     4096;
  7: 1,  628,  25191,  221800,   612295,   592884,   194017,    16384;
  8: 1, 1267,  81853, 1113919,  4835875,  7843369,  4915423,  1101157,   65536;
  9: 1, 2546, 259496, 5274206, 34204646, 85747214, 89311544, 38301794, 6095609, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+3)*x)*P(n-1,x) + (x-x^2)*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[(1 + (n + 3)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {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] = (1 + (n + 3)*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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A144698: triangle, offset 4, from the row P0
%e Example
Triangle begins:
   4: 1;
   5: 1,    4;
   6: 1,   13,     16;
   7: 1,   32,    113,      64;
   8: 1,   71,    531,     821,      256;
   9: 1,  150,   2090,    6470,     5385,     1024;
  10: 1,  309,   7470,   40510,    65745,    33069,     4096;
  11: 1,  628,  25191,  221800,   612295,   592884,   194017,    16384;
  12: 1, 1267,  81853, 1113919,  4835875,  7843369,  4915423,  1101157,   65536;
  13: 1, 2546, 259496, 5274206, 34204646, 85747214, 89311544, 38301794, 6095609, 262144;
%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 >= 5, with P(4,x) = 1.
%t Mathematica
P[4, 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 - 3], {n, 4, 13}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 3)); v[1] = 1; for(n = 5, N, my(i = n - 3); 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 - 3 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 3, v[i])))};
T(13)

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-f86b9381645a3db7
  • 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-f86b9381645a3db7
  • 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-f86b9381645a3db7
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-f86b9381645a3db7

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

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