A178821 → Veronese 2

OEIS matches ?

• A080951: row sums, 15 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:   5;
  2:  15,    15;
  3:  35,   105;
  4:  70,   420,    70;
  5: 126,  1260,   630;
  6: 210,  3150,  3150,   210;
  7: 330,  6930, 11550,  2310;
  8: 495, 13860, 34650, 13860,  495;
  9: 715, 25740, 90090, 60060, 6435;

The row polynomials satisfy ((1/2)*n^2-(1/2)*n)*P(n,x) = (n^2+3*n-4)*P(n-1,x) - ((1/2)*n^2+(7/2)*n+6-((1/2)*n^2+(7/2)*n+6)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5.

P[0, x_] := 1;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n - 4)*P[n - 1, x] - ((1/2)*n^2 + (7/2)*n + 6 - ((1/2)*n^2 + (7/2)*n + 6)*x)*P[n - 2, x])/((1/2)*n^2 - (1/2)*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 3*n - 4)*v[i - 1] - ((1/2)*n^2 + (7/2)*n + 6 - ((1/2)*n^2 + (7/2)*n + 6)*x)*v[i - 2])/((1/2)*n^2 - (1/2)*n)); 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-8ba085c2fb60bb1b
• 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-8ba085c2fb60bb1b
• 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-8ba085c2fb60bb1b
• 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-8ba085c2fb60bb1b

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-8ba085c2fb60bb1b

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

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-20c79d8930f25d5b root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • preserver_image reversal, shift 0 e-12f9ed6b8198f9ce

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-20c79d8930f25d5b (expanded above)

Related sequences ?

Same fingerprint (10)

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

• A028326 → Veronese 2
• A344565 → Veronese 2
• A178820 → Veronese 2
• A007318 → D → Veronese 2
• A038208 → Veronese 2
• A007318 → Veronese 2
• A135838 → Veronese 2
• basis-map coefficients of A113214, shifted → Veronese 2
• A159854 → Veronese 2
• A178822 → 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
4. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003