A008288 → basis A013611 → θ+1

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Submission text ?

Triangle begins:
   1:    1;
   2:    2,     8;
   3:    5,    40,     48;
   4:   12,   144,    384,     256;
   5:   29,   464,   1920,    2816,     1280;
   6:   70,  1400,   7872,   18176,    17920,     6144;
   7:  169,  4056,  28848,   91392,   142080,   104448,    28672;
   8:  408, 11424,  98304,  396800,   849920,   983040,   573440,   131072;
   9:  985, 31520, 318336, 1563136,  4282880,  6832128,  6250496,  3014656,   589824;
  10: 2378, 85608, 992640, 5745152, 19215360, 39284736, 49545216, 37355520, 15335424, 2621440;

The row polynomials satisfy ((6*n-6)*x+(12*n^2-12)*x^2+(8*n^3-8*n)*x^3)*P(n,x) = -(4*n-8+(12*n^2-6*n-48)*x+(8*n^3+28*n^2-108*n-80)*x^2+(24*n^3-56*n^2-160*n-32)*x^3-(64*n^2+64*n)*x^4)*P(n-1,x) + (1+(4*n+10)*x+(4*n^2+36*n+34)*x^2+(32*n^2+104*n+44)*x^3+(72*n^2+104*n+16)*x^4+(32*n^2+32*n)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (12*n^2 - 6*n - 48)*x + (8*n^3 + 28*n^2 - 108*n - 80)*x^2 + (24*n^3 - 56*n^2 - 160*n - 32)*x^3 - (64*n^2 + 64*n)*x^4)*P[n - 1, x] + (1 + (4*n + 10)*x + (4*n^2 + 36*n + 34)*x^2 + (32*n^2 + 104*n + 44)*x^3 + (72*n^2 + 104*n + 16)*x^4 + (32*n^2 + 32*n)*x^5)*D[P[n - 1, x], x])/((6*n - 6)*x + (12*n^2 - 12)*x^2 + (8*n^3 - 8*n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(4*n - 8 + (12*n^2 - 6*n - 48)*x + (8*n^3 + 28*n^2 - 108*n - 80)*x^2 + (24*n^3 - 56*n^2 - 160*n - 32)*x^3 - (64*n^2 + 64*n)*x^4)*v[i - 1] + (1 + (4*n + 10)*x + (4*n^2 + 36*n + 34)*x^2 + (32*n^2 + 104*n + 44)*x^3 + (72*n^2 + 104*n + 16)*x^4 + (32*n^2 + 32*n)*x^5)*deriv(v[i - 1], x))/((6*n - 6)*x + (12*n^2 - 12)*x^2 + (8*n^3 - 8*n)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from 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 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
            • ↑ Liu–Wang theorem with a lag row (above)
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
                • ↑ Liu–Wang theorem with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                  • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
                    no recorded property covers this requirement
                • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                  • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
                    no recorded property covers this requirement
            • ↑ Affine-current basis map theorem (above)
            • ↑ Affine-current basis map theorem (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine-current basis map theorem (above)
        • ↑ Affine-current basis map theorem (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine-current basis map theorem (above)
        • ↑ Affine-current basis map theorem (above)

Related ?

Same fingerprint ?

• delannoy → basis A013611 → 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. [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.