Real-rooted atlas

basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.

palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in [0, ∞) · all roots real, in [−1, 0] · alternating coefficient signs · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30011
400011
5000011
60000011
700000011
8000000011

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-1bf99ad40f056d33 · revision 3 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A026729
derived:mobius_x_over_x_plus_one (equal by e-976e9514cb2529e4)
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 -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): yes

OEIS matches ?

No OEIS matches.

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, 1;
   3: 0, 0, 1;
   4: 0, 0, 0, 1;
   5: 0, 0, 0, 0, 1;
   6: 0, 0, 0, 0, 0, 1;
   7: 0, 0, 0, 0, 0, 0, 1;
   8: 0, 0, 0, 0, 0, 0, 0, 1;
   9: 0, 0, 0, 0, 0, 0, 0, 0, 1;
  10: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*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[x*P[n - 1, 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] = x*v[i - 1]); 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.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-c545fb294b37e535, e-c9bec2c8b266f7d5, e-ef4991d3a529ecb1, e-fcd0089d79cb4b3f
  • Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n(-x) 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-c545fb294b37e535, e-c9bec2c8b266f7d5, e-ef4991d3a529ecb1, e-fcd0089d79cb4b3f
  • 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-6e85c0971e84e272
  • 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-6e85c0971e84e272, e-ad67990681a86223
  • 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-6e85c0971e84e272, e-ad67990681a86223
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-ad67990681a86223

Proof chains

7 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 23 proof steps

Same fingerprint (75)

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