A159854 → x ↦ x+1 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:   1,    4,     4;
  1:   2,   11,    20,     12;
  2:   4,   28,    73,     84,      36;
  3:   8,   68,   230,    387,     324,     108;
  4:  16,  160,   664,   1464,    1809,    1188,     324;
  5:  32,  368,  1808,   4920,    8010,    7803,    4212,     972;
  6:  64,  832,  4720,  15264,   30780,   39636,   31833,   14580,    2916;
  7: 128, 1856, 11936,  44688,  107352,  171612,  182574,  124659,   49572,    8748;
  8: 256, 4096, 29440, 125184,  348768,  665280,  879984,  797040,  473121,  166212,  26244;
  9: 512, 8960, 71168, 338688, 1073088, 2376864, 3755808, 4234032, 3337362, 1751787, 551124, 78732;

The row polynomials satisfy (1+x)*P(n,x) = (2+5*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x + 4*x^2.

P[0, x_] := 1 + 4*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 5*x + 3*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 4*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 5*x + 3*x^2)*v[i - 1])/(1 + x)); 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, 0]
  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)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (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 → x ↦ x+1 → basis A013609
• A391533 → reverse → basis A013610
• A159854 → basis A013609 → 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.