delannoy → basis A013611 → D+1

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Submission text ?

Triangle begins:
  0:     1;
  1:     6,      4;
  2:    25,     52,      16;
  3:    84,    328,     320,       64;
  4:   261,   1512,    2752,     1728,      256;
  5:   770,   5948,   16256,    18880,     8704,     1024;
  6:  2197,  21260,   78160,   136512,   115456,    41984,     4096;
  7:  6120,  71248,  330368,   779136,   989184,   655360,   196608,    16384;
  8: 16745, 227984, 1278464,  3817088,  6550016,  6496256,  3530752,   901120,   65536;
  9: 45182, 704564, 4639744, 16808576, 36580352, 49014784, 39763968, 18300928, 4063232, 262144;

The row polynomials satisfy (4*n^2-2*n+(8*n^3-2*n)*x+(16*n^2-24*n)*x^2+(16*n^2-12*n)*x^3+16*n*x^4)*P(n,x) = -(8*n^3-28*n^2+8+(24*n^3-112*n^2-26*n+70)*x-(32*n^2+116*n-172)*x^2+(48*n^2-100*n+60)*x^3-(16*n+96)*x^4-64*x^5)*P(n-1,x) + (4*n^2-1+(32*n^2+8*n-10)*x+(72*n^2+72*n-24)*x^2+(32*n^2+208*n+44)*x^3+(208*n+228)*x^4+(64*n+224)*x^5+64*x^6)*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[(-(8*n^3 - 28*n^2 + 8 + (24*n^3 - 112*n^2 - 26*n + 70)*x - (32*n^2 + 116*n - 172)*x^2 + (48*n^2 - 100*n + 60)*x^3 - (16*n + 96)*x^4 - 64*x^5)*P[n - 1, x] + (4*n^2 - 1 + (32*n^2 + 8*n - 10)*x + (72*n^2 + 72*n - 24)*x^2 + (32*n^2 + 208*n + 44)*x^3 + (208*n + 228)*x^4 + (64*n + 224)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(4*n^2 - 2*n + (8*n^3 - 2*n)*x + (16*n^2 - 24*n)*x^2 + (16*n^2 - 12*n)*x^3 + 16*n*x^4)]];
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] = (-(8*n^3 - 28*n^2 + 8 + (24*n^3 - 112*n^2 - 26*n + 70)*x - (32*n^2 + 116*n - 172)*x^2 + (48*n^2 - 100*n + 60)*x^3 - (16*n + 96)*x^4 - 64*x^5)*v[i - 1] + (4*n^2 - 1 + (32*n^2 + 8*n - 10)*x + (72*n^2 + 72*n - 24)*x^2 + (32*n^2 + 208*n + 44)*x^3 + (208*n + 228)*x^4 + (64*n + 224)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(4*n^2 - 2*n + (8*n^3 - 2*n)*x + (16*n^2 - 24*n)*x^2 + (16*n^2 - 12*n)*x^3 + 16*n*x^4)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 → D+1

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.