delannoy → basis A013610 → θ+1

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Triangle begins:
  0:    1;
  1:    2,     6;
  2:    5,    30,     27;
  3:   12,   108,    216,     108;
  4:   29,   348,   1080,    1188,     405;
  5:   70,  1050,   4428,    7668,    5670,    1458;
  6:  169,  3042,  16227,   38556,   44955,   24786,    5103;
  7:  408,  8568,  55296,  167400,  268920,  233280,  102060,   17496;
  8:  985, 23640, 179064,  659448, 1355130, 1621296, 1112454,  402408,   59049;
  9: 2378, 64206, 558360, 2423736, 6079860, 9322452, 8817984, 4986360, 1535274, 196830;

The row polynomials satisfy (108*n*x+(162*n^2+324*n)*x^2+(81*n^3+243*n^2+162*n)*x^3)*P(n,x) = -(96*n-96+(216*n^2+324*n-756)*x+(108*n^3+702*n^2-378*n-2052)*x^2+(243*n^3+162*n^2-2025*n-2268)*x^3-(486*n^2+1458*n+972)*x^4)*P(n-1,x) + (32+(96*n+336)*x+(72*n^2+792*n+1332)*x^2+(432*n^2+2268*n+2430)*x^3+(729*n^2+2511*n+1944)*x^4+(243*n^2+729*n+486)*x^5)*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[(-(96*n - 96 + (216*n^2 + 324*n - 756)*x + (108*n^3 + 702*n^2 - 378*n - 2052)*x^2 + (243*n^3 + 162*n^2 - 2025*n - 2268)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*P[n - 1, x] + (32 + (96*n + 336)*x + (72*n^2 + 792*n + 1332)*x^2 + (432*n^2 + 2268*n + 2430)*x^3 + (729*n^2 + 2511*n + 1944)*x^4 + (243*n^2 + 729*n + 486)*x^5)*D[P[n - 1, x], x])/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*n)*x^3)]];
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] = (-(96*n - 96 + (216*n^2 + 324*n - 756)*x + (108*n^3 + 702*n^2 - 378*n - 2052)*x^2 + (243*n^3 + 162*n^2 - 2025*n - 2268)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*v[i - 1] + (32 + (96*n + 336)*x + (72*n^2 + 792*n + 1332)*x^2 + (432*n^2 + 2268*n + 2430)*x^3 + (729*n^2 + 2511*n + 1944)*x^4 + (243*n^2 + 729*n + 486)*x^5)*deriv(v[i - 1], x))/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*n)*x^3)); 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

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 delannoy?
    • Geometric basis map (elementary) polytool ? step ›

      • from delannoy?
        • 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 delannoy?
            • ↑ 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 delannoy?
                • ↑ 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 a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                    no recorded property covers this requirement
                • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                  • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                    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 delannoy?
        • ↑ 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 ?

• A008288 → 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.