basis-map coefficients of A026729, shifted → (θ+1)² → basis A008277

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Triangle begins:
   1:    1;
   2:   -4,     -4;
   3:    9,     27,       9;
   4:  -16,   -112,     -96,      -16;
   5:   25,    375,     625,      250,       25;
   6:  -36,  -1116,   -3240,    -2340,     -540,      -36;
   7:   49,   3087,   14749,    17150,     6860,     1029,      49;
   8:  -64,  -8128,  -61824,  -108864,   -67200,   -17024,   -1792,    -64;
   9:   81,  20655,  245025,   629370,   563031,   214326,   37422,   2916,    81;
  10: -100, -51100, -933000, -3410500, -4252500, -2282700, -588000, -75000, -4500, -100;

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+n^2*x)*P(n-1,x) - n^2*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n^2*x)*P[n - 1, x] - n^2*x*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n^2 + n^2*x)*v[i - 1] - n^2*x*deriv(v[i - 1], x))/(n^2 - 2*n + 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 (−∞, 0]
  by Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= …?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Parent-row provers of polytool (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= …?
        • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= …?
            • ↑ Laguerre’s theorem on multiplier sequences (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= …?
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• A131431 → θ+1 → basis A008277
• A130123 → basis A008277
• A136521 → basis A008277
• A106800: Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
• basis-map coefficients of A026729, shifted → D → basis A008277
• A129186 → basis A008277
• A127648 → D → basis A008277
• A130123 → D → basis A008277
• A048993 → D+1 → reverse
• A132440 → θ+1 → basis A008277
• A136572 → (θ+1)² → basis A008277
• A131436 → basis A008277
• basis-map coefficients of A112227 → (θ+1)² → basis A008277
• A136572 → θ+1 → basis A008277
• image under θ+1 → (θ+1)² → basis A008277
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
• basis-map coefficients of A026729 → D → basis A008277
• basis-map coefficients of A021012 → basis A008277 → reverse
• A130123 → θ+1 → basis A008277
• A132440 → basis A008277
• basis-map coefficients of A112227 → θ+1 → basis A008277
• A136572 → D → basis A008277
• basis-map coefficients of A030528, shifted → D → basis A008277
• basis-map coefficients of A021012 → θ+1 → basis A008277
• A136572 → basis A008277
• basis-map coefficients of A112227 → basis A008277
• basis-map coefficients of A026729, shifted → basis A008277
• image under θ+1 → θ+1 → D → basis A008277
• A048993 → D+1
• image under θ+1 → basis A008277
• basis-map coefficients of A030528, shifted → basis A008277
• A131431 → D → basis A008277
• basis-map coefficients of A026729, shifted → θ+1 → basis A008277
• basis-map coefficients of A030528, shifted → θ+1 → basis A008277
• basis-map coefficients of A021012 → basis A008277
• A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
• A130123 → (θ+1)² → basis A008277
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A008277
• A134309 → basis A008277
• A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
• basis-map coefficients of A026729 → (θ+1)² → basis A008277
• A131431 → basis A008277
• A130123 → basis A008277 → reverse
• basis-map coefficients of A112227 → basis A008277 → reverse
• A048993 → reverse

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.