A023531 → Veronese 2

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

The row polynomials satisfy P(n,x) = 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[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] = 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-0e4f0d097e1dbf06, e-8742693abd852f1e
• 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-0e4f0d097e1dbf06, e-8742693abd852f1e
• 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-0e4f0d097e1dbf06, e-8742693abd852f1e
• Real-rooted. Every row P_n, n >= 1, 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-fc5f49d4b55941b1
• Coefficients of one sign. Every nonzero row, n >= 1: 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-fc5f49d4b55941b1
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-fc5f49d4b55941b1

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

• veronese_section r=2, j=0, shift 0 e-0e4f0d097e1dbf06

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

  • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d

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

      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
    • parent_provers periodic_product e-b7eca039219a7f53 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-7e44b66e04c497d6

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
        • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
• veronese_section r=2, j=0, shift 0 e-8742693abd852f1e

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

  • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-5a4f04ff49b735c1 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-4adca88d3fcf673e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-5a4f04ff49b735c1 (expanded above)
• veronese_section r=2, j=1, shift 0 e-fc5f49d4b55941b1

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

  • premise (sequence): basis-map coefficients of A scaled Hermite triangle.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-470643e70af29063 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-f9cce4eb6656235e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of A scaled Hermite triangle.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-470643e70af29063 (expanded above)

Related sequences ?

Same fingerprint (16)

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

• A136572 → Veronese 2 (j=1)
• basis-map coefficients of A112227 → Veronese 2
• A136572 → Veronese 2
• basis-map coefficients of A026729, shifted → Veronese 2 (j=1)
• basis-map coefficients of A030528, shifted → Veronese 2 (j=1)
• basis-map coefficients of A026729 → Veronese 2 (j=1)
• A130123 → Veronese 2 (j=1)
• basis-map coefficients of A021012 → Veronese 2 (j=1)
• A010054 → reverse → Veronese 2
• basis-map coefficients of A021012 → Veronese 2
• A130123 → Veronese 2
• A010054 → reverse → Veronese 2
• basis-map coefficients of A026729, shifted → Veronese 2
• A023531 → D → Veronese 2 (j=1)
• A133080 → Veronese 2 (j=1)
• A010054 → reverse → 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