A143491 → basis A007318 → D+1

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Submission text ?

Triangle begins:
  0:         19,           9,          1;
  1:        107,          71,         15,          1;
  2:        702,         580,        173,         22,         1;
  3:       5274,        5104,       1910,        345,        30,         1;
  4:      44712,       48860,      21559,       4915,       610,        39,        1;
  5:     422568,      509004,     255045,      68929,     10850,       994,       49,       1;
  6:    4407120,     5753736,    3193392,     984704,    184569,     21532,     1526,      60,      1;
  7:   50292720,    70290936,   42461556,   14562764,   3126732,    435981,    39480,    2238,     72,    1;
  8:  623471040,   924118272,  599807916,  224709140,  53850635,   8630433,   937083,   68070,   3165,   85,  1;
  9: 8344624320, 13020978816, 8991695196, 3630938036, 953466855, 171076499, 21412545, 1870143, 111705, 4345, 99, 1;

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(2*n+6)*x+x^2)*P(n-2,x) + (2*n+7+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 19 + 9*x + x^2, P(1,x) = 107 + 71*x + 15*x^2 + x^3.

P[0, x_] := 19 + 9*x + x^2;
P[1, x_] := 107 + 71*x + 15*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (2*n + 6)*x + x^2)*P[n - 2, x] + (2*n + 7 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 19 + 9*x + x^2; v[2] = 107 + 71*x + 15*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (2*n + 6)*x + x^2)*v[i - 2] + (2*n + 7 + 2*x)*v[i - 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A136124 → x ↦ x+1 → D+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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.