A132393 → Veronese 2

OEIS matches ?

• A188588: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A001710: row sums, 13 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;
  1: 0;
  2: 0,      1;
  3: 0,      3;
  4: 0,     11,     1;
  5: 0,     50,    10;
  6: 0,    274,    85,    1;
  7: 0,   1764,   735,   21;
  8: 0,  13068,  6769,  322,  1;
  9: 0, 109584, 67284, 4536, 36;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 0; for(n = 2, N, my(i = n + 1); v[i] = (2*n - 3)*v[i - 1] - (n^2 - 4*n + 4 - x)*v[i - 2]); 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-b3060fae442da909
• 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-b3060fae442da909
• 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-b3060fae442da909

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-b3060fae442da909

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

  • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A130534 → Veronese 2 (j=1)

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