A062196 → reverse → (θ+1)²

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Triangle begins:
  0:  1;
  1:  3,    4;
  2:  6,   32,      9;
  3: 10,  120,    135,     16;
  4: 15,  320,    810,    384,      25;
  5: 21,  700,   3150,   3360,     875,      36;
  6: 28, 1344,   9450,  17920,   10500,    1728,     49;
  7: 36, 2352,  23814,  70560,   73500,   27216,   3087,     64;
  8: 45, 3840,  52920, 225792,  367500,  241920,  61740,   5120,   81;
  9: 55, 5940, 106920, 620928, 1455300, 1496880, 679140, 126720, 8019, 100;

The row polynomials satisfy (n^2+2*n+(2*n^4+2*n^3-4*n^2)*x+(n^6+2*n^5)*x^2)*P(n,x) = (n^2+4*n+4+(2*n^4+6*n^3+12*n^2-8)*x+(n^6+4*n^5+13*n^4+6*n^3-11*n^2-8*n+4)*x^2+(3*n^6+10*n^5+9*n^4-4*n^3-6*n^2)*x^3)*P(n-1,x) + ((2*n+2)*x+(4*n^3+6*n^2-4)*x^2+(2*n^5+8*n^4+8*n^3-2*n+2)*x^3-(2*n^5+8*n^4+12*n^3+6*n^2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 4 + (2*n^4 + 6*n^3 + 12*n^2 - 8)*x + (n^6 + 4*n^5 + 13*n^4 + 6*n^3 - 11*n^2 - 8*n + 4)*x^2 + (3*n^6 + 10*n^5 + 9*n^4 - 4*n^3 - 6*n^2)*x^3)*P[n - 1, x] + ((2*n + 2)*x + (4*n^3 + 6*n^2 - 4)*x^2 + (2*n^5 + 8*n^4 + 8*n^3 - 2*n + 2)*x^3 - (2*n^5 + 8*n^4 + 12*n^3 + 6*n^2)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n + (2*n^4 + 2*n^3 - 4*n^2)*x + (n^6 + 2*n^5)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 4 + (2*n^4 + 6*n^3 + 12*n^2 - 8)*x + (n^6 + 4*n^5 + 13*n^4 + 6*n^3 - 11*n^2 - 8*n + 4)*x^2 + (3*n^6 + 10*n^5 + 9*n^4 - 4*n^3 - 6*n^2)*x^3)*v[i - 1] + ((2*n + 2)*x + (4*n^3 + 6*n^2 - 4)*x^2 + (2*n^5 + 8*n^4 + 8*n^3 - 2*n + 2)*x^3 - (2*n^5 + 8*n^4 + 12*n^3 + 6*n^2)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n + (2*n^4 + 2*n^3 - 4*n^2)*x + (n^6 + 2*n^5)*x^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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from reversal of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.