basis-map coefficients of A112227 → basis A008277 → reverse

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Triangle begins:
   1: 1,  3,    1;
   2: 1,  6,    7,     1;
   3: 1, 10,   25,    15,      1;
   4: 1, 15,   65,    90,     31,      1;
   5: 1, 21,  140,   350,    301,     63,       1;
   6: 1, 28,  266,  1050,   1701,    966,     127,       1;
   7: 1, 36,  462,  2646,   6951,   7770,    3025,     255,      1;
   8: 1, 45,  750,  5880,  22827,  42525,   34105,    9330,    511,     1;
   9: 1, 55, 1155, 11880,  63987, 179487,  246730,  145750,  28501,  1023,    1;
  10: 1, 66, 1705, 22275, 159027, 627396, 1323652, 1379400, 611501, 86526, 2047, 1;

The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 3*x + x^2.

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

P(N) = {my(v = vector(N)); v[1] = 1 + 3*x + x^2; for(n = 2, N, my(i = n); v[i] = (1 + (n + 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); 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 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 basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • 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 A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

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

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.