basis-map coefficients of A112227 → θ+1 → basis A008277

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
   1:  3,     9,       3;
   2:  4,    28,      24,       4;
   3:  5,    75,     125,      50,        5;
   4:  6,   186,     540,     390,       90,        6;
   5:  7,   441,    2107,    2450,      980,      147,       7;
   6:  8,  1016,    7728,   13608,     8400,     2128,     224,       8;
   7:  9,  2295,   27225,   69930,    62559,    23814,    4158,     324,      9;
   8: 10,  5110,   93300,  341050,   425250,   228270,   58800,    7500,    450,    10;
   9: 11, 11253,  313511, 1603250,  2714030,  1974357,  703857,  130680,  12705,   605,  11;
  10: 12, 24564, 1038312, 7338012, 16552800, 15883824, 7528752, 1908324, 267300, 20460, 792, 12;

The row polynomials satisfy (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) = 3 + 9*x + 3*x^2.

P[1, x_] := 3 + 9*x + 3*x^2;
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 + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 3 + 9*x + 3*x^2; 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 + 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 of basis-map coefficients of A scaled Hermite triangle.?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? 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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one of basis-map coefficients of A scaled Hermite triangle.?
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

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

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.