A135838 → Veronese 2 (j=1)

OEIS matches ?

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:   0;
  1:   2;
  2:   4;
  3:  12,    4;
  4:  16,   16;
  5:  40,   80,    8;
  6:  48,  160,   48;
  7: 112,  560,  336,   16;
  8: 128,  896,  896,  128;
  9: 288, 2688, 4032, 1152, 32;

The row polynomials satisfy P(n,x) = (4+4*x)*P(n-2,x) - (4-8*x+4*x^2)*P(n-4,x) for n >= 4, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 4, P(3,x) = 12 + 4*x.

P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 4;
P[3, x_] := 12 + 4*x;
P[n_, x_] := P[n, x] = Expand[(4 + 4*x)*P[n - 2, x] - (4 - 8*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] = 0; v[2] = 2; v[3] = 4; v[4] = 12 + 4*x; for(n = 4, N, my(i = n + 1); v[i] = (4 + 4*x)*v[i - 2] - (4 - 8*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-b898a1f1e7c8e42a
• 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-b898a1f1e7c8e42a
• 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-b898a1f1e7c8e42a

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=1, shift 0 e-b898a1f1e7c8e42a

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

  • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-07f7d31230240b98

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-51cb02553cf4a8ac root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-51cb02553cf4a8ac (expanded above)

Related sequences ?

Same fingerprint (11)

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

• A159854 → Veronese 2 (j=1)
• A038208 → Veronese 2 (j=1)
• A007318 → D → Veronese 2 (j=1)
• A344565 → Veronese 2 (j=1)
• A178822 → Veronese 2 (j=1)
• A178821 → Veronese 2 (j=1)
• A007318 → Veronese 2 (j=1)
• A028326 → Veronese 2 (j=1)
• A178820 → Veronese 2 (j=1)
• basis-map coefficients of A113214, shifted → Veronese 2 (j=1)
• A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.

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