A159854 → basis A013609 → reverse

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Submission text ?

Triangle begins:
  0:    4,     4,     1;
  1:    8,    16,    10,      2;
  2:   16,    48,    52,     24,      4;
  3:   32,   128,   200,    152,     56,      8;
  4:   64,   320,   656,    704,    416,    128,     16;
  5:  128,   768,  1952,   2720,   2240,   1088,    288,     32;
  6:  256,  1792,  5440,   9344,   9920,   6656,   2752,    640,     64;
  7:  512,  4096, 14464,  29568,  38528,  33152,  18816,   6784,   1408,   128;
  8: 1024,  9216, 37120,  88064, 136192, 143360, 103936,  51200,  16384,  3072,  256;
  9: 2048, 20480, 92672, 250368, 448512, 559104, 494592, 310272, 135168, 38912, 6656, 512;

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

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

• A167194 → basis A007318
• A159854 → basis A013609
• A391533 → reverse
• A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.

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.