A159854 → basis A013609 → basis A007318

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Submission text ?

Triangle begins:
  0:       9,       12,        4;
  1:      36,       66,       40,        8;
  2:     144,      336,      292,      112,       16;
  3:     576,     1632,     1840,     1032,      288,       32;
  4:    2304,     7680,    10624,     7808,     3216,      704,       64;
  5:    9216,    35328,    57856,    52480,    28480,     9248,     1664,     128;
  6:   36864,   159744,   302080,   325632,   218880,    93952,    25152,    3840,     256;
  7:  147456,   712704,  1527808,  1906688,  1526784,   813568,   288512,   65664,    8704,    512;
  8:  589824,  3145728,  7536640, 10682368,  9920512,  6307840,  2781184,  839680,  166144,  19456,  1024;
  9: 2359296, 13762560, 36438016, 57802752, 61046784, 45072384, 23740416, 8921088, 2343936, 410112, 43008, 2048;

The row polynomials satisfy P(n,x) = (4+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 12*x + 4*x^2.

P[0, x_] := 9 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 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] = 9 + 12*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + 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 (−∞, -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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (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_A013609 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_A013609 of Riordan array (1-x,x/(1-x)).?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A391533 → x ↦ x/(x+1)
• A391533 → reverse → 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.