Real-rooted atlas

A008292: Eulerian

A008292 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
\(P_{n}(x) = \left(1 - x + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31416
411111124
512666261120
6157302302571720
7112011912416119112015040
81247429315619156194293247140320

Show all 15 stored rows

Generated b-files: A008292, A008292 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-ff270bcb802a85af · revision 2 · JSON
polytool family
eulerian
Definition source
named_family:eulerian
Fingerprint
f6-605cb6decd8257f5

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: 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 ?

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,    4,     1;
   4: 1,   11,    11,      1;
   5: 1,   26,    66,     26,       1;
   6: 1,   57,   302,    302,      57,       1;
   7: 1,  120,  1191,   2416,    1191,     120,      1;
   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;
   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;
  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;
%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.

A008292: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    4,     1;
   4: 1,   11,    11,      1;
   5: 1,   26,    66,     26,       1;
   6: 1,   57,   302,    302,      57,       1;
   7: 1,  120,  1191,   2416,    1191,     120,      1;
   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;
   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;
  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;
%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.

A008292: reversed triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    4,     1;
   4: 1,   11,    11,      1;
   5: 1,   26,    66,     26,       1;
   6: 1,   57,   302,    302,      57,       1;
   7: 1,  120,  1191,   2416,    1191,     120,      1;
   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;
   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;
  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where 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[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 - 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 = 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 >= 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-ab3cdcfae289b48d
  • 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-6c46c948823f0a8c, e-ab3cdcfae289b48d, e-c61ecf4ef1c45372
  • 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-6c46c948823f0a8c, e-ab3cdcfae289b48d, e-c61ecf4ef1c45372
  • 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-ab3cdcfae289b48d
  • 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-6c46c948823f0a8c, e-c61ecf4ef1c45372

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 15 proof steps

Same fingerprint (6)

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