A125662 → Veronese 2 (j=1)

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.

Triangle begins:
  0:     0;
  1:     1;
  2:     6;
  3:    25,     1;
  4:    90,    12;
  5:   300,    86,    1;
  6:   954,   480,   18;
  7:  2939,  2305,  183,   1;
  8:  8850, 10008, 1386,  24;
  9: 26195, 40426, 8715, 316, 1;

The row polynomials satisfy P(n,x) = 6*P(n-1,x) - (11-x)*P(n-2,x) + 6*P(n-3,x) - P(n-4,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 6, P(3,x) = 25 + x.

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 6;
P[3, x_] := 25 + x;
P[n_, x_] := P[n, x] = Expand[6*P[n - 1, x] - (11 - x)*P[n - 2, x] + 6*P[n - 3, x] - P[n - 4, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 6; v[4] = 25 + x; for(n = 4, N, my(i = n + 1); v[i] = 6*v[i - 1] - (11 - x)*v[i - 2] + 6*v[i - 3] - v[i - 4]); 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)

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-34c8c2d842a569b5
• 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-34c8c2d842a569b5
• 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-34c8c2d842a569b5
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-34c8c2d842a569b5

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• veronese_section r=2, j=1, shift 0 e-34c8c2d842a569b5

  Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?

  • premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-af8f298ca8f41adf root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • theorem_check three_term_recurrence e-c7df247a48c8552e root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

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. [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
4. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023