Real-rooted atlas

A111577: Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.

A111577 nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A111577
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31517
412112135
5185105221214
613418203253511523
7113656081407078051112349
8154614393246781142101596701112052

Show all 15 stored rows

Generated b-files: A003575 (row sums), A111577 ?

Computed in your browser ?

Technical details
Node id
n-cab14672969f6507 · revision 1 · JSON
Definition source
oeis_catalog:A111577
Fingerprint
f6-c48676221a04fdaf

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: 1,     1;
   3: 1,     5,       1;
   4: 1,    21,      12,       1;
   5: 1,    85,     105,      22,       1;
   6: 1,   341,     820,     325,      35,      1;
   7: 1,  1365,    6081,    4070,     780,     51,     1;
   8: 1,  5461,   43932,   46781,   14210,   1596,    70,    1;
   9: 1, 21845,  312985,  511742,  231511,  39746,  2926,   92,   1;
  10: 1, 87381, 2212740, 5430405, 3521385, 867447, 95340, 4950, 117, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 3*x*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 + x)*P[n - 1, x] + 3*x*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 + x)*v[i - 1] + 3*x*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.

A111577: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,     1;
   3: 1,     5,       1;
   4: 1,    21,      12,       1;
   5: 1,    85,     105,      22,       1;
   6: 1,   341,     820,     325,      35,      1;
   7: 1,  1365,    6081,    4070,     780,     51,     1;
   8: 1,  5461,   43932,   46781,   14210,   1596,    70,    1;
   9: 1, 21845,  312985,  511742,  231511,  39746,  2926,   92,   1;
  10: 1, 87381, 2212740, 5430405, 3521385, 867447, 95340, 4950, 117, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 3*x*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 + x)*P[n - 1, x] + 3*x*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 + x)*v[i - 1] + 3*x*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.

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-94f85f79154d7ed6
  • 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-94f85f79154d7ed6
  • 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-94f85f79154d7ed6
  • 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-94f85f79154d7ed6
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 1] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 1] with m = 0, target (-inf, 0]
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-e9721368474f6fa1

Proof chains

2 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 2 proof steps

No candidate equalities or related sequences.

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