Real-rooted atlas

A131436: Triangle read by rows, (n-1) zeros followed by 2^n - 1.

A131436 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A131436

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1033
20077
30001515
400003131
5000006363
6000000127127
70000000255255
800000000511511
900000000010231023
10000000000020472047
110000000000040954095
1200000000000081918191
1300000000000001638316383
14000000000000003276732767

b-files: A126646 (row sums), A131436, A168604 (row sums) ?

Technical details
Node id
n-6394a556f7ad3cd4 · revision 0 · JSON
Definition source
oeis_catalog:A131436
Fingerprint
f6-edf6954a4cd8adfc

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 ?

  • A131436 triangle 80 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A255047, A126646, A153010, A168604, A239678, A088041, A060152, A000225

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 3;
  2: 0, 0, 7;
  3: 0, 0, 0, 15;
  4: 0, 0, 0,  0, 31;
  5: 0, 0, 0,  0,  0, 63;
  6: 0, 0, 0,  0,  0,  0, 127;
  7: 0, 0, 0,  0,  0,  0,   0, 255;
  8: 0, 0, 0,  0,  0,  0,   0,   0, 511;
  9: 0, 0, 0,  0,  0,  0,   0,   0,   0, 1023;
%F Formula
The row polynomials satisfy P(n,x) = 3*x*P(n-1,x) - 2*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3*x;
P[n_, x_] := P[n, x] = Expand[3*x*P[n - 1, x] - 2*x^2*P[n - 2, x]];
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] = 3*x; for(n = 2, N, my(i = n + 1); v[i] = 3*x*v[i - 1] - 2*x^2*v[i - 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

A131436 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 0, 3;
   3: 0, 0, 7;
   4: 0, 0, 0, 15;
   5: 0, 0, 0,  0, 31;
   6: 0, 0, 0,  0,  0, 63;
   7: 0, 0, 0,  0,  0,  0, 127;
   8: 0, 0, 0,  0,  0,  0,   0, 255;
   9: 0, 0, 0,  0,  0,  0,   0,   0, 511;
  10: 0, 0, 0,  0,  0,  0,   0,   0,   0, 1023;
%F Formula
The row polynomials satisfy P(n,x) = 3*x*P(n-1,x) - 2*x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3*x;
P[n_, x_] := P[n, x] = Expand[3*x*P[n - 1, x] - 2*x^2*P[n - 2, 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; v[2] = 3*x; for(n = 3, N, my(i = n); v[i] = 3*x*v[i - 1] - 2*x^2*v[i - 2]); 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

  • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Same fingerprint ?

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