Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → reverse

nonnegative coefficients

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
215511
3110231549
41197610952257
51362245315442031539
616963122043641287687710299
7113417498471200892520816113414075905

Show all 15 stored rows

Generated b-files: A059340 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-4438653f9e73a43c · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-629d13292f83019a

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,   2;
  2: 1,   5,     5;
  3: 1,  10,    23,     15;
  4: 1,  19,    76,    109,     52;
  5: 1,  36,   224,    531,    544,     203;
  6: 1,  69,   631,   2204,   3641,    2876,     877;
  7: 1, 134,  1749,   8471,  20089,   25208,   16113,    4140;
  8: 1, 263,  4838,  31331, 100171,  177631,  178564,   95495,  21147;
  9: 1, 520, 13454, 113807, 471827, 1109473, 1559788, 1302451, 597155, 115975;
%F Formula
The row polynomials satisfy P(n,x) = (1-(n-3)*x-(n-1)*x^2)*P(n-1,x) + (x+2*x^2+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[(1 - (n - 3)*x - (n - 1)*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*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 - (n - 1)*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*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.

A059340: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:      1;
   2:      2,      1;
   3:      5,      5,       1;
   4:     15,     23,      10,       1;
   5:     52,    109,      76,      19,       1;
   6:    203,    544,     531,     224,      36,      1;
   7:    877,   2876,    3641,    2204,     631,     69,      1;
   8:   4140,  16113,   25208,   20089,    8471,   1749,    134,     1;
   9:  21147,  95495,  178564,  177631,  100171,  31331,   4838,   263,   1;
  10: 115975, 597155, 1302451, 1559788, 1109473, 471827, 113807, 13454, 520, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1-(n-4)*x-(n-2)*x^2)*P(n-1,x) + (x+2*x^2+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[(1 - (n - 4)*x - (n - 2)*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];
Table[Reverse[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 - 4)*x - (n - 2)*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(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 >= 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-0bec24e5cc539db1
  • 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-0bec24e5cc539db1
  • 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-0bec24e5cc539db1

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 (5)

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