A257609 → θ+1

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Submission text ?

Triangle begins:
  0:   1;
  1:   2,       4;
  2:   4,      32,       12;
  3:   8,     176,      264,        32;
  4:  16,     832,     3168,      1664,         80;
  5:  32,    3648,    28992,     38656,       9120,        192;
  6:  64,   15360,   228672,    618496,     381120,      46080,        448;
  7: 128,   63232,  1648512,   7996928,    9996160,    3297024,     221312,      1024;
  8: 256,  257024, 11218944,  90351616,  199923200,  135527424,   26177536,   1028096,    2304;
  9: 512, 1037312, 73482240, 932233216, 3354506240, 4025407488, 1631408128, 195952640, 4667904, 5120;

The row polynomials satisfy P(n,x) = -(4*n^2-4)*x^2*P(n-2,x) - ((4*n+4)*x^2-(4*n+4)*x^3)*d/dx P(n-2,x) + (2+4*n*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 4)*x^2*P[n - 2, x] - ((4*n + 4)*x^2 - (4*n + 4)*x^3)*D[P[n - 2, x], x] + (2 + 4*n*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 4)*x^2*v[i - 2] - ((4*n + 4)*x^2 - (4*n + 4)*x^3)*deriv(v[i - 2], x) + (2 + 4*n*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Parent-row provers of polytool polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • Parent-row provers of polytool polytool ? step › direct check
    • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A334218 → D
• A173018 → θ+1
• A257620 → θ+1
• A008292 → θ+1
• A008292 → θ+1 (shift +2) → reverse
• row-factor cofactor of A101842 → θ+1
• A173018 → reverse (shift +1) → reverse → θ+1
• A008292 → θ+1 (shift +2)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.