A008288 → basis A013610 → θ+1

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Triangle begins:
   1:    1;
   2:    2,     6;
   3:    5,    30,     27;
   4:   12,   108,    216,     108;
   5:   29,   348,   1080,    1188,     405;
   6:   70,  1050,   4428,    7668,    5670,    1458;
   7:  169,  3042,  16227,   38556,   44955,   24786,    5103;
   8:  408,  8568,  55296,  167400,  268920,  233280,  102060,   17496;
   9:  985, 23640, 179064,  659448, 1355130, 1621296, 1112454,  402408,   59049;
  10: 2378, 64206, 558360, 2423736, 6079860, 9322452, 8817984, 4986360, 1535274, 196830;

The row polynomials satisfy ((108*n-108)*x+(162*n^2-162)*x^2+(81*n^3-81*n)*x^3)*P(n,x) = -(96*n-192+(216*n^2-108*n-864)*x+(108*n^3+378*n^2-1458*n-1080)*x^2+(243*n^3-567*n^2-1620*n-324)*x^3-(486*n^2+486*n)*x^4)*P(n-1,x) + (32+(96*n+240)*x+(72*n^2+648*n+612)*x^2+(432*n^2+1404*n+594)*x^3+(729*n^2+1053*n+162)*x^4+(243*n^2+243*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[(-(96*n - 192 + (216*n^2 - 108*n - 864)*x + (108*n^3 + 378*n^2 - 1458*n - 1080)*x^2 + (243*n^3 - 567*n^2 - 1620*n - 324)*x^3 - (486*n^2 + 486*n)*x^4)*P[n - 1, x] + (32 + (96*n + 240)*x + (72*n^2 + 648*n + 612)*x^2 + (432*n^2 + 1404*n + 594)*x^3 + (729*n^2 + 1053*n + 162)*x^4 + (243*n^2 + 243*n)*x^5)*D[P[n - 1, x], x])/((108*n - 108)*x + (162*n^2 - 162)*x^2 + (81*n^3 - 81*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] = (-(96*n - 192 + (216*n^2 - 108*n - 864)*x + (108*n^3 + 378*n^2 - 1458*n - 1080)*x^2 + (243*n^3 - 567*n^2 - 1620*n - 324)*x^3 - (486*n^2 + 486*n)*x^4)*v[i - 1] + (32 + (96*n + 240)*x + (72*n^2 + 648*n + 612)*x^2 + (432*n^2 + 1404*n + 594)*x^3 + (729*n^2 + 1053*n + 162)*x^4 + (243*n^2 + 243*n)*x^5)*deriv(v[i - 1], x))/((108*n - 108)*x + (162*n^2 - 162)*x^2 + (81*n^3 - 81*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_A013610 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 A013610 → 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.