basis-map coefficients of A030528, shifted → D → basis A008277

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Triangle begins:
  0:  0;
  1: -1;
  2:  2,     2;
  3: -3,    -9,     -3;
  4:  4,    28,     24,      4;
  5: -5,   -75,   -125,    -50,     -5;
  6:  6,   186,    540,    390,     90,      6;
  7: -7,  -441,  -2107,  -2450,   -980,   -147,    -7;
  8:  8,  1016,   7728,  13608,   8400,   2128,   224,    8;
  9: -9, -2295, -27225, -69930, -62559, -23814, -4158, -324, -9;

The row polynomials satisfy (n-1)*P(n,x) = -(n+n*x)*P(n-1,x) - n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = -1.

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = -1; for(n = 2, N, my(i = n + 1); v[i] = (-(n + n*x)*v[i - 1] - n*x*deriv(v[i - 1], x))/(n - 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 (−∞, 0]
  by Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 derivative of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

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

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.