A159854 → x ↦ x+1 → basis A013610

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Submission text ?

Triangle begins:
  0:     4,     12,       9;
  1:    12,     48,      63,       27;
  2:    36,    180,     333,      270,       81;
  3:   108,    648,    1539,     1809,     1053,      243;
  4:   324,   2268,    6561,    10044,     8586,     3888,      729;
  5:   972,   7776,   26487,    49815,    55890,    37422,    13851,     2187;
  6:  2916,  26244,  102789,   228906,   317115,   279936,   153819,    48114,     6561;
  7:  8748,  87480,  387099,   995085,  1638063,  1791153,  1301265,   605799,   164025,   19683;
  8: 26244, 288684, 1423737,  4146552,  7899444, 10287648,  9277254,  5721192,  2309472,  551124,   59049;
  9: 78732, 944784, 5137263, 16710867, 36137988, 54561276, 58694706, 44995338, 24091992, 8581788, 1830519, 177147;

The row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 12*x + 9*x^2.

P[0, x_] := 4 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 4 + 12*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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 (−∞, 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 shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
    • Affine, reversal and Möbius substitutions (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
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A391533 → basis A013609 → reverse
• A391533 → basis A013609

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.