A056242 → Veronese 2 → x ↦ x+1

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Triangle begins:
   1:     1;
   2:     1;
   3:     5,     4;
   4:    17,    16;
   5:    58,    73,    16;
   6:   198,   309,   112;
   7:   676,  1259,   648,    64;
   8:  2308,  4995,  3328,   640;
   9:  7880, 19432, 15889,  4592,  256;
  10: 26904, 74464, 72153, 27920, 3328;

The row polynomials satisfy P(n,x) = x*P(n-4,x) - 4*x*P(n-3,x) - (2-4*x)*P(n-2,x) + 4*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 5 + 4*x, P(4,x) = 17 + 16*x.

P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 5 + 4*x;
P[4, x_] := 17 + 16*x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 4, x] - 4*x*P[n - 3, x] - (2 - 4*x)*P[n - 2, x] + 4*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 5 + 4*x; v[4] = 17 + 16*x; for(n = 5, N, my(i = n); v[i] = x*v[i - 4] - 4*x*v[i - 3] - (2 - 4*x)*v[i - 2] + 4*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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_0 of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 …?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
        • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            no recorded property covers this requirement
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
                • ↑ Affine-current basis map theorem (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
                • ↑ Affine-current basis map theorem (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