Real-rooted atlas

A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.

A010054 → reverse

A127773 palindromic γ-positive nonnegative coefficients only rational roots

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

Recurrence from OEIS A127773
\(P_{n}(x) = \left(3\,x\right)\,P_{n-1}(x) + \left(-3\,x^{2}\right)\,P_{n-2}(x) + \left(x^{3}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\,x\)
\(P_{2}(x) = 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1033
20066
30001010
400001515
5000002121
60000002828
700000003636

Show all 15 stored rows

Generated b-files: A127773 ?

Computed in your browser ?

Other recurrences

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

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(x^{3} + 3\,x^{4} + 3\,x^{5} + x^{6}\right)\,P_{n-3}(x) + \left(-3\,x^{2} - 9\,x^{3} - 9\,x^{4} - 3\,x^{5}\right)\,P_{n-2}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\,x\)
\(P_{2}(x) = 6\,x^{2}\)
Technical details
Node id
n-511a2225fa2191c9 · revision 2 · JSON
Definition source
oeis_catalog:A127773
derived:mobius_x_over_x_plus_one (equal by e-f20366b40a82ef63)
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

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 3;
  2: 0, 0, 6;
  3: 0, 0, 0, 10;
  4: 0, 0, 0,  0, 15;
  5: 0, 0, 0,  0,  0, 21;
  6: 0, 0, 0,  0,  0,  0, 28;
  7: 0, 0, 0,  0,  0,  0,  0, 36;
  8: 0, 0, 0,  0,  0,  0,  0,  0, 45;
  9: 0, 0, 0,  0,  0,  0,  0,  0,  0, 55;
%F Formula
The row polynomials satisfy P(n,x) = 3*x*P(n-1,x) - 3*x^2*P(n-2,x) + x^3*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3*x, P(2,x) = 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3*x;
P[2, x_] := 6*x^2;
P[n_, x_] := P[n, x] = Expand[3*x*P[n - 1, x] - 3*x^2*P[n - 2, x] + x^3*P[n - 3, 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; v[3] = 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = 3*x*v[i - 1] - 3*x^2*v[i - 2] + x^3*v[i - 3]); 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)

Generated with polytool c48629374597.

A127773: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 0, 3;
   3: 0, 0, 6;
   4: 0, 0, 0, 10;
   5: 0, 0, 0,  0, 15;
   6: 0, 0, 0,  0,  0, 21;
   7: 0, 0, 0,  0,  0,  0, 28;
   8: 0, 0, 0,  0,  0,  0,  0, 36;
   9: 0, 0, 0,  0,  0,  0,  0,  0, 45;
  10: 0, 0, 0,  0,  0,  0,  0,  0,  0, 55;
%F Formula
The row polynomials satisfy P(n,x) = 3*x*P(n-1,x) - 3*x^2*P(n-2,x) + x^3*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3*x, P(3,x) = 6*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3*x;
P[3, x_] := 6*x^2;
P[n_, x_] := P[n, x] = Expand[3*x*P[n - 1, x] - 3*x^2*P[n - 2, x] + x^3*P[n - 3, 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; v[3] = 6*x^2; for(n = 4, N, my(i = n); v[i] = 3*x*v[i - 1] - 3*x^2*v[i - 2] + x^3*v[i - 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, 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-9c0dd75b125bab92
  • Coefficients of one sign. Every nonzero row, n >= 0: 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-9c0dd75b125bab92, e-a72a60f6dc2c6e62
  • Hadamard kernel (Malo). Rows P_n, n >= 0, 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-9c0dd75b125bab92, e-a72a60f6dc2c6e62
  • Real-rooted. Every row P_n, n >= 0, 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-a72a60f6dc2c6e62

Proof chains

3 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 7 proof steps

Same fingerprint (75)

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

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  5. [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
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.