delannoy → basis A013611 → θ+1

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Triangle begins:
  0:    1;
  1:    2,     8;
  2:    5,    40,     48;
  3:   12,   144,    384,     256;
  4:   29,   464,   1920,    2816,     1280;
  5:   70,  1400,   7872,   18176,    17920,     6144;
  6:  169,  4056,  28848,   91392,   142080,   104448,    28672;
  7:  408, 11424,  98304,  396800,   849920,   983040,   573440,   131072;
  8:  985, 31520, 318336, 1563136,  4282880,  6832128,  6250496,  3014656,   589824;
  9: 2378, 85608, 992640, 5745152, 19215360, 39284736, 49545216, 37355520, 15335424, 2621440;

The row polynomials satisfy (6*n*x+(12*n^2+24*n)*x^2+(8*n^3+24*n^2+16*n)*x^3)*P(n,x) = -(4*n-4+(12*n^2+18*n-42)*x+(8*n^3+52*n^2-28*n-152)*x^2+(24*n^3+16*n^2-200*n-224)*x^3-(64*n^2+192*n+128)*x^4)*P(n-1,x) + (1+(4*n+14)*x+(4*n^2+44*n+74)*x^2+(32*n^2+168*n+180)*x^3+(72*n^2+248*n+192)*x^4+(32*n^2+96*n+64)*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 + (12*n^2 + 18*n - 42)*x + (8*n^3 + 52*n^2 - 28*n - 152)*x^2 + (24*n^3 + 16*n^2 - 200*n - 224)*x^3 - (64*n^2 + 192*n + 128)*x^4)*P[n - 1, x] + (1 + (4*n + 14)*x + (4*n^2 + 44*n + 74)*x^2 + (32*n^2 + 168*n + 180)*x^3 + (72*n^2 + 248*n + 192)*x^4 + (32*n^2 + 96*n + 64)*x^5)*D[P[n - 1, x], x])/(6*n*x + (12*n^2 + 24*n)*x^2 + (8*n^3 + 24*n^2 + 16*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 + (12*n^2 + 18*n - 42)*x + (8*n^3 + 52*n^2 - 28*n - 152)*x^2 + (24*n^3 + 16*n^2 - 200*n - 224)*x^3 - (64*n^2 + 192*n + 128)*x^4)*v[i - 1] + (1 + (4*n + 14)*x + (4*n^2 + 44*n + 74)*x^2 + (32*n^2 + 168*n + 180)*x^3 + (72*n^2 + 248*n + 192)*x^4 + (32*n^2 + 96*n + 64)*x^5)*deriv(v[i - 1], x))/(6*n*x + (12*n^2 + 24*n)*x^2 + (8*n^3 + 24*n^2 + 16*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_A013611 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 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.