A159854 → x ↦ x+1 → basis A013609

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Submission text ?

Triangle begins:
  0:     4,      8,       4;
  1:    12,     32,      28,       8;
  2:    36,    120,     148,      80,      16;
  3:   108,    432,     684,     536,     208,      32;
  4:   324,   1512,    2916,    2976,    1696,     512,      64;
  5:   972,   5184,   11772,   14760,   11040,    4928,    1216,     128;
  6:  2916,  17496,   45684,   67824,   62640,   36864,   13504,    2816,    256;
  7:  8748,  58320,  172044,  294840,  323568,  235872,  114240,   35456,   6400,    512;
  8: 26244, 192456,  632772, 1228608, 1560384, 1354752,  814464,  334848,  90112,  14336,  1024;
  9: 78732, 629856, 2283228, 4951368, 7138368, 7185024, 5152896, 2633472, 940032, 223232, 31744, 2048;

The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 8*x + 4*x^2.

P[0, x_] := 4 + 8*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 4 + 8*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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 (−∞, 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 A013610
• A024462 → 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.