A008288 → basis A007318 → θ+1

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Submission text ?

Triangle begins:
   1:    1;
   2:    2,     2;
   3:    5,    10,     3;
   4:   12,    36,    24,     4;
   5:   29,   116,   120,    44,     5;
   6:   70,   350,   492,   284,    70,     6;
   7:  169,  1014,  1803,  1428,   555,   102,     7;
   8:  408,  2856,  6144,  6200,  3320,   960,   140,    8;
   9:  985,  7880, 19896, 24424, 16730,  6672,  1526,  184,   9;
  10: 2378, 21402, 62040, 89768, 75060, 38364, 12096, 2280, 234, 10;

The row polynomials satisfy ((12*n-12)*x+(6*n^2-6)*x^2+(n^3-n)*x^3)*P(n,x) = -(32*n-64+(24*n^2-12*n-96)*x+(4*n^3+14*n^2-54*n-40)*x^2+(3*n^3-7*n^2-20*n-4)*x^3-(2*n^2+2*n)*x^4)*P(n-1,x) + (32+(32*n+80)*x+(8*n^2+72*n+68)*x^2+(16*n^2+52*n+22)*x^3+(9*n^2+13*n+2)*x^4+(n^2+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[(-(32*n - 64 + (24*n^2 - 12*n - 96)*x + (4*n^3 + 14*n^2 - 54*n - 40)*x^2 + (3*n^3 - 7*n^2 - 20*n - 4)*x^3 - (2*n^2 + 2*n)*x^4)*P[n - 1, x] + (32 + (32*n + 80)*x + (8*n^2 + 72*n + 68)*x^2 + (16*n^2 + 52*n + 22)*x^3 + (9*n^2 + 13*n + 2)*x^4 + (n^2 + n)*x^5)*D[P[n - 1, x], x])/((12*n - 12)*x + (6*n^2 - 6)*x^2 + (n^3 - 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] = (-(32*n - 64 + (24*n^2 - 12*n - 96)*x + (4*n^3 + 14*n^2 - 54*n - 40)*x^2 + (3*n^3 - 7*n^2 - 20*n - 4)*x^3 - (2*n^2 + 2*n)*x^4)*v[i - 1] + (32 + (32*n + 80)*x + (8*n^2 + 72*n + 68)*x^2 + (16*n^2 + 52*n + 22)*x^3 + (9*n^2 + 13*n + 2)*x^4 + (n^2 + n)*x^5)*deriv(v[i - 1], x))/((12*n - 12)*x + (6*n^2 - 6)*x^2 + (n^3 - 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 shift_x_plus_one of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
    • 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 [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)
    • 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) (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)
    • 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 → x ↦ x+1 → 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.