A200536 → basis A007318 → reverse

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Submission text ?

Triangle begins:
  0:   1;
  1:   2,     7,      6;
  2:   4,    28,     73,      84,       36;
  3:   8,    84,    366,     847,     1098,      756,       216;
  4:  16,   224,   1368,    4760,    10321,    14280,     12312,      6048,       1296;
  5:  32,   560,   4400,   20440,    62170,   129367,    186510,    183960,     118800,      45360,       7776;
  6:  64,  1344,  12912,   75040,   293820,   816564,   1651609,   2449692,    2644380,    2026080,    1045872,     326592,      46656;
  7: 128,  3136,  35616,  248528,  1190392,  4140108,  10782086,  21360031,   32346258,   37260972,   32140584,   20130768,    8654688,    2286144,     279936;
  8: 256,  7168,  93952,  765184,  4334176, 18104128,  57687280, 143035312,  278905249,  429105936,  519185520,  488811456,  351068256,  185939712,   68491008,   15676416,   1679616;
  9: 512, 16128, 239616, 2231040, 14588352, 71138592, 268108512, 798506352, 1905181362, 3668760487, 5715544086, 7186557168, 7238929824, 5762225952, 3544969536, 1626428160, 524040192, 105815808, 10077696;

The row polynomials satisfy P(n,x) = (2+7*x+6*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + 7*x + 6*x^2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + 7*x + 6*x^2)*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_A007318 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
        • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A200536 → basis A007318
• A200536 → basis A013610 → reverse
• A200536 → 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.