delannoy → basis A013609 → θ+1

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Triangle begins:
  0:    1;
  1:    2,     4;
  2:    5,    20,     12;
  3:   12,    72,     96,     32;
  4:   29,   232,    480,    352,      80;
  5:   70,   700,   1968,   2272,    1120,     192;
  6:  169,  2028,   7212,  11424,    8880,    3264,    448;
  7:  408,  5712,  24576,  49600,   53120,   30720,   8960,   1024;
  8:  985, 15760,  79584, 195392,  267680,  213504,  97664,  23552,  2304;
  9: 2378, 42804, 248160, 718144, 1200960, 1227648, 774144, 291840, 59904, 5120;

The row polynomials satisfy (3*n*x+(3*n^2+6*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(4*n-4+(6*n^2+9*n-21)*x+(2*n^3+13*n^2-7*n-38)*x^2+(3*n^3+2*n^2-25*n-28)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (2+(4*n+14)*x+(2*n^2+22*n+37)*x^2+(8*n^2+42*n+45)*x^3+(9*n^2+31*n+24)*x^4+(2*n^2+6*n+4)*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[(-(4*n - 4 + (6*n^2 + 9*n - 21)*x + (2*n^3 + 13*n^2 - 7*n - 38)*x^2 + (3*n^3 + 2*n^2 - 25*n - 28)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (2 + (4*n + 14)*x + (2*n^2 + 22*n + 37)*x^2 + (8*n^2 + 42*n + 45)*x^3 + (9*n^2 + 31*n + 24)*x^4 + (2*n^2 + 6*n + 4)*x^5)*D[P[n - 1, x], x])/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*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] = (-(4*n - 4 + (6*n^2 + 9*n - 21)*x + (2*n^3 + 13*n^2 - 7*n - 38)*x^2 + (3*n^3 + 2*n^2 - 25*n - 28)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (2 + (4*n + 14)*x + (2*n^2 + 22*n + 37)*x^2 + (8*n^2 + 42*n + 45)*x^3 + (9*n^2 + 31*n + 24)*x^4 + (2*n^2 + 6*n + 4)*x^5)*deriv(v[i - 1], x))/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*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_A013609 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 A013609 → 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.