A081579 → γ → (θ+1)²

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Triangle begins:
  0: 1;
  1: 1;
  2: 1,  16;
  3: 1,  32;
  4: 1,  48,  144;
  5: 1,  64,  432;
  6: 1,  80,  864,  1024;
  7: 1,  96, 1440,  4096;
  8: 1, 112, 2160, 10240,  6400;
  9: 1, 128, 3024, 20480, 32000;

The row polynomials satisfy (n^2+n+(12*n^3-8*n)*x+(16*n^4+16*n^3)*x^2)*P(n,x) = (n^2+n+(12*n^3+32*n^2-8*n-32)*x+(16*n^4+176*n^3+128*n^2-224*n-64)*x^2+(128*n^4+512*n^3+384*n^2-512*n-512)*x^3)*P(n-1,x) - ((n+1)*x+(12*n^2+40*n+24)*x^2+(16*n^3+272*n^2+512*n+192)*x^3+(256*n^3+1280*n^2+2048*n+1024)*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 + n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (16*n^4 + 176*n^3 + 128*n^2 - 224*n - 64)*x^2 + (128*n^4 + 512*n^3 + 384*n^2 - 512*n - 512)*x^3)*P[n - 1, x] - ((n + 1)*x + (12*n^2 + 40*n + 24)*x^2 + (16*n^3 + 272*n^2 + 512*n + 192)*x^3 + (256*n^3 + 1280*n^2 + 2048*n + 1024)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (12*n^3 - 8*n)*x + (16*n^4 + 16*n^3)*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 + n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (16*n^4 + 176*n^3 + 128*n^2 - 224*n - 64)*x^2 + (128*n^4 + 512*n^3 + 384*n^2 - 512*n - 512)*x^3)*v[i - 1] - ((n + 1)*x + (12*n^2 + 40*n + 24)*x^2 + (16*n^3 + 272*n^2 + 512*n + 192)*x^3 + (256*n^3 + 1280*n^2 + 2048*n + 1024)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (12*n^3 - 8*n)*x + (16*n^4 + 16*n^3)*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

Proof chains ? JSON

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

  • from gamma_polynomial of Pascal-(1,4,1) array.?
    • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

      • from Pascal-(1,4,1) array.?
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Pascal-(1,4,1) array.?
            • ↑ Liu–Wang theorem with a lag row (above)
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Pascal-(1,4,1) array.?
                • ↑ Liu–Wang theorem with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (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.