A143491 → basis A013609 → basis A013609

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Submission text ?

Triangle begins:
  0:          30,           44,           16;
  1:         210,          428,          288,           64;
  2:        1680,         4264,         4016,         1664,          256;
  3:       15120,        45096,        53200,        31040,         8960,         1024;
  4:      151200,       511440,       712384,       523200,       213760,        46080,         4096;
  5:     1663200,      6230640,      9881984,      8604736,      4444160,      1361920,       229376,       16384;
  6:    19958400,     81420480,    143506368,    142784768,     87748864,     34119680,      8200192,     1114112,       65536;
  7:   259459200,   1138299840,   2191264704,   2430227456,   1711874304,    794551296,    243081216,    47284224,     5308416,     262144;
  8:  3632428800,  16974034560,  35230905216,  42788243200,  33687150080,  17971215360,   6581342208,  1634304000,   263454720,   24903680,   1048576;
  9: 54486432000, 269140233600, 596359716480, 782747268864, 676460224000, 404316830720, 170604994560, 50839928832, 10489036800, 1427374080, 115343360, 4194304;

The row polynomials satisfy P(n,x) = (n+6+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 30 + 44*x + 16*x^2.

P[0, x_] := 30 + 44*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 6 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 30 + 44*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 6 + 4*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 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)

Related ?

Same fingerprint ?

• A132393 → basis A013611 → 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. [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.