A152198 → Veronese 2

OEIS matches ?

• A158780: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A016116: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A131572: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A163403: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A060546: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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.

Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1;
  4: 1, 1;
  5: 1, 1;
  6: 1, 3;
  7: 1, 3;
  8: 1, 6, 1;
  9: 1, 6, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = 2*v[i - 2] - (1 - x)*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-c2e81851a0a2b89d
• 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-c2e81851a0a2b89d
• 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-c2e81851a0a2b89d

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-c2e81851a0a2b89d

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

  • premise (sequence): Triangle read by rows, A007318 rows repeated.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-3ca43d0d97d5e532

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-35a33ce2a6b89415 root: theorem check

          Parent-row provers of polytool polytool ?

    • gamma_polynomial scale 1, shift 0 e-78c582ff5f7b6a70

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-6e3ca670df3d0a74 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-b2e3690a28625635

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-6e3ca670df3d0a74 (expanded above)
    • parent_provers periodic_product e-b274e5cc6dbb8c2d 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.

• A274496 → Veronese 2

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