A272867 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:        0;
  1:        6,         2;
  2:       72,        96,         36,          4;
  3:      648,      1512,       1296,        504,          90,           6;
  4:     5184,     17280,      23328,      16416,        6480,        1440,         168,          8;
  5:    38880,    168480,     311040,     319680,      200880,       79920,       20160,       3120,        270,         10;
  6:   279936,   1492992,    3499200,    4743360,     4121280,     2405376,      961632,     263520,      48600,       5760,       396,        12;
  7:  1959552,  12410496,   35271936,   59439744,    66134880,    51220512,    28377216,   11382336,    3306744,     687960,     99792,      9576,      546,      14;
  8: 13436928,  98537472,  329204736,  663634944,   901393920,   872653824,   621395712,  331174656,  133156224,   40400640,   9180864,   1536192,   183456,   14784,    720,    16;
  9: 90699264, 755827200, 2902376448, 6812522496, 10934300160, 12721411584, 11098902528, 7411585536, 3833303616, 1544080320, 484475904, 117790848, 21936096, 3066336, 311040, 21600, 918, 18;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 6*n*x + n*x^2)*v[i - 1])/(n - 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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)

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.