A064861 → Veronese 2

OEIS matches ?

• A026565: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 1,   2;
  1: 1,   8;
  2: 1,  13,    4;
  3: 1,  25,   28;
  4: 1,  33,   66,    8;
  5: 1,  51,  192,   80;
  6: 1,  62,  321,  248,   16;
  7: 1,  86,  681,  968,  208;
  8: 1, 100,  985, 1970,  800,  32;
  9: 1, 130, 1765, 5336, 3920, 512;

The row polynomials satisfy P(n,x) = (2+4*x)*P(n-2,x) - (1-5*x+4*x^2)*P(n-4,x) for n >= 4, with P(0,x) = 1 + 2*x, P(1,x) = 1 + 8*x, P(2,x) = 1 + 13*x + 4*x^2, P(3,x) = 1 + 25*x + 28*x^2.

P[0, x_] := 1 + 2*x;
P[1, x_] := 1 + 8*x;
P[2, x_] := 1 + 13*x + 4*x^2;
P[3, x_] := 1 + 25*x + 28*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x)*P[n - 2, x] - (1 - 5*x + 4*x^2)*P[n - 4, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; v[2] = 1 + 8*x; v[3] = 1 + 13*x + 4*x^2; v[4] = 1 + 25*x + 28*x^2; for(n = 4, N, my(i = n + 1); v[i] = (2 + 4*x)*v[i - 2] - (1 - 5*x + 4*x^2)*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-4995ac39f8921064
• 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-4995ac39f8921064
• 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-4995ac39f8921064

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=0, shift 0 e-4995ac39f8921064

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

  • premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-bbffc3f85cdc52e2 root: theorem check

      Parent-row provers of polytool 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