A113413 → Veronese 2 (j=1) → x ↦ x+1

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Submission text ?

Triangle begins:
  0:    0;
  1:    1;
  2:    4;
  3:    9,    1;
  4:   20,    8;
  5:   49,   34,    1;
  6:  120,  112,   12;
  7:  289,  339,   75,   1;
  8:  696,  992,  340,  16;
  9: 1681, 2820, 1302, 132, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-4,x) + (2+2*x)*P(n-3,x) + x*P(n-2,x) + 2*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4, P(3,x) = 9 + x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 4; v[4] = 9 + x; for(n = 4, N, my(i = n + 1); v[i] = (1 + x)*v[i - 4] + (2 + 2*x)*v[i - 3] + x*v[i - 2] + 2*v[i - 1]); 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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from veronese_section_2_1 of A Riordan array of coordination sequences.?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from A Riordan array of coordination sequences.?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of A Riordan array of coordination sequences.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from A Riordan array of coordination sequences.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from A Riordan array of coordination sequences.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)

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