basis-map coefficients of A112227 → basis A008277 → x ↦ x+1

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Submission text ?

Triangle begins:
   1:       5,        5,        1;
   2:      15,       22,        9,        1;
   3:      52,       99,       61,       14,        1;
   4:     203,      471,      385,      135,       20,       1;
   5:     877,     2386,     2416,     1140,      260,      27,       1;
   6:    4140,    12867,    15470,     9156,     2835,     455,      35,      1;
   7:   21147,    73681,   102215,    72590,    28441,    6230,     742,     44,     1;
   8:  115975,   446620,   700311,   578929,   274386,   76503,   12474,   1146,    54,    1;
   9:  678570,  2856457,  4984651,  4692500,  2607760,  884751,  184317,  23220,  1695,   65,  1;
  10: 4213597, 19217243, 36872561, 38878191, 24762815, 9906919, 2521827, 406857, 40755, 2420, 77, 1;

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 5 + 5*x + x^2.

P[1, x_] := 5 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 5 + 5*x + x^2; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + (1 + x)*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 (−∞, -1]
  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)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by 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 ?

• A137597 → reverse
• basis-map coefficients of A021012 → basis A008277 → x ↦ x+1
• A008278 → x ↦ x/(x+1)
• A137597: Triangle read by rows: A008277 * A007318.
• A008277 → x ↦ x+1
• A008277 → x ↦ x+1 → reverse
• A130123 → basis A008277 → x ↦ x+1

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.