A272867 → x ↦ x+1 → reverse

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0: 1;
  1: 1,  6,    6;
  2: 1, 12,   48,    72,     36;
  3: 1, 18,  126,   432,    756,     648,     216;
  4: 1, 24,  240,  1296,   4104,    7776,    8640,     5184,      1296;
  5: 1, 30,  390,  2880,  13320,   40176,   79920,   103680,     84240,     38880,      7776;
  6: 1, 36,  576,  5400,  32940,  137376,  400896,   824256,   1185840,   1166400,    746496,     279936,      46656;
  7: 1, 42,  798,  9072,  68796,  367416, 1422792,  4053888,   8536752,  13226976,  14859936,   11757312,    6205248,    1959552,     279936;
  8: 1, 48, 1056, 14112, 128016,  834624, 4040064, 14795136,  41396832,  88770816, 145442304,  180278784,  165908736,  109734912,   49268736,  13436928,   1679616;
  9: 1, 54, 1350, 20736, 219024, 1687392, 9815904, 44043264, 154408032, 425922624, 926448192, 1585557504, 2120235264, 2186860032, 1703130624, 967458816, 377913600, 90699264, 10077696;

The row polynomials satisfy P(n,x) = (1+6*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[(1 + 6*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] = (1 + 6*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, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
        • 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, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
            • ↑ Iterated real-rootedness preservers (above)
            • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Candidate equalities ?

• A272867 → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A272867 → x ↦ x+1

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.