A159854 → basis A013609 → basis A013610

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Submission text ?

Triangle begins:
  0:       9,       36,        36;
  1:      36,      198,       360,        216;
  2:     144,     1008,      2628,       3024,       1296;
  3:     576,     4896,     16560,      27864,      23328,        7776;
  4:    2304,    23040,     95616,     210816,     260496,      171072,       46656;
  5:    9216,   105984,    520704,    1416960,    2306880,     2247264,     1213056,      279936;
  6:   36864,   479232,   2718720,    8792064,   17729280,    22830336,    18335808,     8398080,     1679616;
  7:  147456,  2138112,  13750272,   51480576,  123669504,   197697024,   210325248,   143607168,    57106944,   10077696;
  8:  589824,  9437184,  67829760,  288423936,  803561472,  1532805120,  2027483136,  1836380160,  1090070784,  382952448,   60466176;
  9: 2359296, 41287680, 327942144, 1560674304, 4944789504, 10952589312, 17306763264, 19510419456, 15378564096, 8072234496, 2539579392, 362797056;

The row polynomials satisfy P(n,x) = (4+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 36*x + 36*x^2.

P[0, x_] := 9 + 36*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 6*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 + 36*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + 6*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)

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

Related ?

Same fingerprint ?

• A391533 → x ↦ x+1 → basis A013609
• A159854 → x ↦ x+1 → x ↦ x/(x+1)
• A391533 → reverse → basis A013610

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.