A091044 → basis A013611 → θ+1

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Triangle begins:
  0:      1;
  1:      4,      16;
  2:     16,     128,       144;
  3:     64,     768,      1920,      1024;
  4:    256,    4096,     16128,     20480,       6400;
  5:   1024,   20480,    110592,    229376,     179200,      36864;
  6:   4096,   98304,    675840,   1966080,    2580480,    1376256,     200704;
  7:  16384,  458752,   3833856,  14417920,   27033600,   24772608,    9633792,    1048576;
  8:  65536, 2097152,  20643840,  95420416,  234291200,  311427072,  211943424,   62914560,   5308416;
  9: 262144, 9437184, 106954752, 587202560, 1789132800, 3148873728, 3148873728, 1660944384, 389283840, 26214400;

The row polynomials satisfy (2*n^2+n+(4*n^3+2*n^2)*x)*P(n,x) = (8*n^2+4*n+(16*n^3+56*n^2+8*n-8)*x+(48*n^3+72*n^2-16*n-8)*x^2)*P(n-1,x) - ((4*n+2)*x+(8*n^2+32*n+12)*x^2+(32*n^2+64*n+16)*x^3)*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[((8*n^2 + 4*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (48*n^3 + 72*n^2 - 16*n - 8)*x^2)*P[n - 1, x] - ((4*n + 2)*x + (8*n^2 + 32*n + 12)*x^2 + (32*n^2 + 64*n + 16)*x^3)*D[P[n - 1, x], x])/(2*n^2 + n + (4*n^3 + 2*n^2)*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] = ((8*n^2 + 4*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (48*n^3 + 72*n^2 - 16*n - 8)*x^2)*v[i - 1] - ((4*n + 2)*x + (8*n^2 + 32*n + 12)*x^2 + (32*n^2 + 64*n + 16)*x^3)*deriv(v[i - 1], x))/(2*n^2 + n + (4*n^3 + 2*n^2)*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 basis_map_A013611 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
    • Geometric basis map (elementary) polytool ? step ›

      • from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
            • ↑ Three-term recurrence criterion (above)
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A172431 → θ+1
• A078812 → reverse → θ+1

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
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.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023