A159854 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0: 1,  2,   1;
  1: 1,  4,   5,   2;
  2: 1,  6,  13,  12,    4;
  3: 1,  8,  25,  38,   28,    8;
  4: 1, 10,  41,  88,  104,   64,    16;
  5: 1, 12,  61, 170,  280,  272,   144,    32;
  6: 1, 14,  85, 292,  620,  832,   688,   320,    64;
  7: 1, 16, 113, 462, 1204, 2072,  2352,  1696,   704,  128;
  8: 1, 18, 145, 688, 2128, 4480,  6496,  6400,  4096, 1536,  256;
  9: 1, 20, 181, 978, 3504, 8736, 15456, 19392, 16896, 9728, 3328, 512;

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

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

• A159854 → x ↦ x+1
• A167194 → 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.