Real-rooted atlas

A099097: Riordan array (1, 3+x).

A099097 nonnegative coefficients

Proved for all n: all roots real, in (−4/9, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A099097
\(P_{n}(x) = \left(3\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2033
301910
40062733
50012781109
60009108243360
70001544057291189
8000012270145821873927
9000019012155103656112970
1000000155405103174961968342837
110000011352835204125904959049141481
12000000189451360878732196830177147467280
1300000011895670612362952456495395314411543321
140000000211512306182624401082565212576415943235097243
150000000125210206153090108256538972346908733478296916835050

Generated b-files: A020704 (row sums), A099097, A304249 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Basis-map coefficients of A099097, shifted · proved equal for all n (proof step)

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(4 + 9\,x - 2\,n\right)\,P_{n-1}(x) + \left(4\,x + 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-5d39164c4108e043 · revision 1 · JSON
Definition source
oeis_catalog:A099097
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A099097 (equal by e-b70532453d2b0e11)
Fingerprint
f6-22d7f6f9cdb525db

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: 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 3;
   3: 0, 1, 9;
   4: 0, 0, 6, 27;
   5: 0, 0, 1, 27,  81;
   6: 0, 0, 0,  9, 108, 243;
   7: 0, 0, 0,  1,  54, 405,  729;
   8: 0, 0, 0,  0,  12, 270, 1458, 2187;
   9: 0, 0, 0,  0,   1,  90, 1215, 5103,  6561;
  10: 0, 0, 0,  0,   0,  15,  540, 5103, 17496, 19683;
%F Formula
The row polynomials satisfy P(n,x) = 3*x*P(n-1,x) + x*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] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {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] + x*v[i - 2]); 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.

A099097: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 0, 3;
  2: 0, 1, 9;
  3: 0, 0, 6, 27;
  4: 0, 0, 1, 27,  81;
  5: 0, 0, 0,  9, 108, 243;
  6: 0, 0, 0,  1,  54, 405,  729;
  7: 0, 0, 0,  0,  12, 270, 1458, 2187;
  8: 0, 0, 0,  0,   1,  90, 1215, 5103,  6561;
  9: 0, 0, 0,  0,   0,  15,  540, 5103, 17496, 19683;
%F Formula
The row polynomials satisfy P(n,x) = 3*x*P(n-1,x) + x*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] + x*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A304249: reversed triangle, offset 0, from the row P1, low zeros dropped
%e Example
Triangle begins:
  0:     1;
  1:     3;
  2:     9,     1;
  3:    27,     6;
  4:    81,    27,    1;
  5:   243,   108,    9;
  6:   729,   405,   54,   1;
  7:  2187,  1458,  270,  12;
  8:  6561,  5103, 1215,  90,  1;
  9: 19683, 17496, 5103, 540, 15;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = 3*x*P(n-1,x) + x*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] + x*P[n - 2, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], 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] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Proof chains

15 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 26 proof steps

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

References

  1. [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