Real-rooted atlas

A128540: Triangle A127647 * A097806, read by rows.

A128540 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence from A128540

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
20224
300336
40005510
500008816
600000131326
7000000212142

All 15 rows b-files: A078642 (row sums), A128540, A128588 (row sums) ?

Technical details
Node id
n-a009f9b98df3ca1d · revision 0 · JSON
Definition source
oeis_catalog:A128540
Fingerprint
f6-abab48ed12ce8f73

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: 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): yes

Matches ?

  • A128540 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A128588, A068922, A198834, A163733, A078642, A055389, A006355, A118658

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1, 1;
  2: 0, 2, 2;
  3: 0, 0, 3, 3;
  4: 0, 0, 0, 5, 5;
  5: 0, 0, 0, 0, 8,  8;
  6: 0, 0, 0, 0, 0, 13, 13;
  7: 0, 0, 0, 0, 0,  0, 21, 21;
  8: 0, 0, 0, 0, 0,  0,  0, 34, 34;
  9: 0, 0, 0, 0, 0,  0,  0,  0, 55, 55;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2)*x*P(n-1,x) + (n-2)*x^2*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 2*x + 2*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 2*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*x*P[n - 1, x] + (n - 2)*x^2*P[n - 2, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 2*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2)*x*v[i - 1] + (n - 2)*x^2*v[i - 2])/(n - 2)); 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)

polytool c48629374597

A128540 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1, 1;
   3: 0, 2, 2;
   4: 0, 0, 3, 3;
   5: 0, 0, 0, 5, 5;
   6: 0, 0, 0, 0, 8,  8;
   7: 0, 0, 0, 0, 0, 13, 13;
   8: 0, 0, 0, 0, 0,  0, 21, 21;
   9: 0, 0, 0, 0, 0,  0,  0, 34, 34;
  10: 0, 0, 0, 0, 0,  0,  0,  0, 55, 55;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = (n-3)*x*P(n-1,x) + (n-3)*x^2*P(n-2,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 2*x + 2*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 2*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3)*x*P[n - 1, x] + (n - 3)*x^2*P[n - 2, x])/(n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 2*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3)*x*v[i - 1] + (n - 3)*x^2*v[i - 2])/(n - 3)); 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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.