A391533 → basis A013609 → D+1

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Submission text ?

Triangle begins:
  0:    21,     20,       4;
  1:    60,    106,      56,       8;
  2:   156,    416,     388,     144,      16;
  3:   384,   1384,    1904,    1224,     352,       32;
  4:   912,   4160,    7648,    7168,    3536,      832,      64;
  5:  2112,  11680,   27008,   33600,   24000,     9632,    1920,     128;
  6:  4800,  31232,   87232,  135936,  128320,    74240,   25152,    4352,     256;
  7: 10752,  80512,  263936,  495488,  584192,   445312,  216832,   63616,    9728,    512;
  8: 23808, 201728,  759808, 1671168, 2369024,  2250752, 1440768,  606208,  156928,  21504,  1024;
  9: 52224, 494080, 2103296, 5308416, 8804352, 10042368, 7999488, 4417536, 1637376, 379392, 47104, 2048;

The row polynomials satisfy (3*n+4+(2*n+7)*x+2*x^2)*P(n,x) = (6*n+14+(10*n+32)*x+(4*n+22)*x^2+4*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 21 + 20*x + 4*x^2.

P[0, x_] := 21 + 20*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 14 + (10*n + 32)*x + (4*n + 22)*x^2 + 4*x^3)*P[n - 1, x])/(3*n + 4 + (2*n + 7)*x + 2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 21 + 20*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 14 + (10*n + 32)*x + (4*n + 22)*x^2 + 4*x^3)*v[i - 1])/(3*n + 4 + (2*n + 7)*x + 2*x^2)); 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013609 of 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.?
    • Geometric basis map (elementary) polytool ? step ›

      • from 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.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of 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.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from 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.?
                • ↑ Liu–Wang theorem (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.