A249252 → basis A007318 → D+1

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Submission text ?

Triangle begins:
  0:      1;
  1:      9,       4;
  2:     41,      40,       8;
  3:    185,     284,     128,       16;
  4:    793,    1680,    1224,      352,      32;
  5:   3305,    8964,    9120,     4272,     896,      64;
  6:  13481,   44664,   58632,    38592,   13216,    2176,     128;
  7:  54105,  212012,  341760,   291536,  140672,   37824,    5120,    256;
  8: 214393,  970592, 1857544,  1952288, 1223840,  464128,  102528,  11776,   512;
  9: 840841, 4320340, 9580192, 11985136, 9275776, 4575808, 1425920, 267008, 26624, 1024;

The row polynomials satisfy (4*n^3+17*n^2-34*n+5+(8*n^3+54*n^2-59*n-33)*x+(44*n^2-2*n-74)*x^2+(8*n^2+28*n-44)*x^3+(8*n-8)*x^4)*P(n,x) = -(28*n^3+67*n^2-18*n-5+(24*n^3+154*n^2+91*n+33)*x+(116*n^2+218*n+74)*x^2+(24*n^2+132*n+44)*x^3+(24*n+8)*x^4)*P(n-1,x) + (68*n^2+357*n+408+(148*n^2+1117*n+1619)*x+(96*n^2+1312*n+2575)*x^2+(16*n^2+712*n+2084)*x^3+(176*n+896)*x^4+(16*n+192)*x^5+16*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(28*n^3 + 67*n^2 - 18*n - 5 + (24*n^3 + 154*n^2 + 91*n + 33)*x + (116*n^2 + 218*n + 74)*x^2 + (24*n^2 + 132*n + 44)*x^3 + (24*n + 8)*x^4)*P[n - 1, x] + (68*n^2 + 357*n + 408 + (148*n^2 + 1117*n + 1619)*x + (96*n^2 + 1312*n + 2575)*x^2 + (16*n^2 + 712*n + 2084)*x^3 + (176*n + 896)*x^4 + (16*n + 192)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(4*n^3 + 17*n^2 - 34*n + 5 + (8*n^3 + 54*n^2 - 59*n - 33)*x + (44*n^2 - 2*n - 74)*x^2 + (8*n^2 + 28*n - 44)*x^3 + (8*n - 8)*x^4)]];
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] = (-(28*n^3 + 67*n^2 - 18*n - 5 + (24*n^3 + 154*n^2 + 91*n + 33)*x + (116*n^2 + 218*n + 74)*x^2 + (24*n^2 + 132*n + 44)*x^3 + (24*n + 8)*x^4)*v[i - 1] + (68*n^2 + 357*n + 408 + (148*n^2 + 1117*n + 1619)*x + (96*n^2 + 1312*n + 2575)*x^2 + (16*n^2 + 712*n + 2084)*x^3 + (176*n + 896)*x^4 + (16*n + 192)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(4*n^3 + 17*n^2 - 34*n + 5 + (8*n^3 + 54*n^2 - 59*n - 33)*x + (44*n^2 - 2*n - 74)*x^2 + (8*n^2 + 28*n - 44)*x^3 + (8*n - 8)*x^4)); 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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.