A159854 → basis A013610 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:       16,        24,         9;
  1:       80,       168,       117,         27;
  2:      400,      1080,      1089,        486,         81;
  3:     2000,      6600,      8685,       5697,       1863,        243;
  4:    10000,     39000,     63225,      54540,      26406,       6804,       729;
  5:    50000,    225000,    433125,     462375,     295650,     113238,     24057,      2187;
  6:   250000,   1275000,   2840625,    3611250,    2865375,    1453140,    459999,     83106,      6561;
  7:  1250000,   7125000,  18028125,   26578125,   25160625,   15861825,   6659415,   1795527,    282123,    19683;
  8:  6250000,  39375000, 111515625,  186975000,  205537500,  154791000,  80882550,  28955880,   6797196,   944784,   59049;
  9: 31250000, 215625000, 675703125, 1269421875, 1588612500, 1390567500, 868785750, 387427050, 120853620, 25115508, 3129597, 177147;

The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + 24*x + 9*x^2.

P[0, x_] := 16 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 16 + 24*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 3*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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Riordan array (1-x,x/(1-x)).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • 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
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of Riordan array (1-x,x/(1-x)).?
    • ↑ Geometric basis map (elementary) (above)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.