A008288 → basis A013609 → D+1

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Submission text ?

Triangle begins:
   1:     1;
   2:     4,      2;
   3:    15,     18,      4;
   4:    48,    100,     56,       8;
   5:   145,    436,    424,     152,      16;
   6:   420,   1662,   2360,    1464,     384,     32;
   7:  1183,   5822,  10972,    9960,    4496,    928,     64;
   8:  3264,  19240,  45392,   54896,   36224,  12800,   2176,   128;
   9:  8865,  60936, 173072,  262992,  231456, 119296,  34560,  4992,   256;
  10: 23780, 186842, 621328, 1140304, 1263232, 868160, 365952, 89728, 11264, 512;

The row polynomials satisfy (2*n^2-4*n+2+(2*n^3-4*n^2+2*n)*x+(6*n^2-16*n+10)*x^2+(2*n^2-n-1)*x^3+(2*n-2)*x^4)*P(n,x) = -(4*n^3-22*n^2+24*n+(6*n^3-32*n^2+12*n+36)*x+(10*n^2-58*n+60)*x^2+(6*n^2-17*n-14)*x^3+(2*n-22)*x^4-4*x^5)*P(n-1,x) + (4*n^2-4*n+(16*n^2-4*n-6)*x+(18*n^2+34*n-14)*x^2+(4*n^2+66*n+17)*x^3+(30*n+55)*x^4+(4*n+28)*x^5+4*x^6)*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[(-(4*n^3 - 22*n^2 + 24*n + (6*n^3 - 32*n^2 + 12*n + 36)*x + (10*n^2 - 58*n + 60)*x^2 + (6*n^2 - 17*n - 14)*x^3 + (2*n - 22)*x^4 - 4*x^5)*P[n - 1, x] + (4*n^2 - 4*n + (16*n^2 - 4*n - 6)*x + (18*n^2 + 34*n - 14)*x^2 + (4*n^2 + 66*n + 17)*x^3 + (30*n + 55)*x^4 + (4*n + 28)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(2*n^2 - 4*n + 2 + (2*n^3 - 4*n^2 + 2*n)*x + (6*n^2 - 16*n + 10)*x^2 + (2*n^2 - n - 1)*x^3 + (2*n - 2)*x^4)]];
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] = (-(4*n^3 - 22*n^2 + 24*n + (6*n^3 - 32*n^2 + 12*n + 36)*x + (10*n^2 - 58*n + 60)*x^2 + (6*n^2 - 17*n - 14)*x^3 + (2*n - 22)*x^4 - 4*x^5)*v[i - 1] + (4*n^2 - 4*n + (16*n^2 - 4*n - 6)*x + (18*n^2 + 34*n - 14)*x^2 + (4*n^2 + 66*n + 17)*x^3 + (30*n + 55)*x^4 + (4*n + 28)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(2*n^2 - 4*n + 2 + (2*n^3 - 4*n^2 + 2*n)*x + (6*n^2 - 16*n + 10)*x^2 + (2*n^2 - n - 1)*x^3 + (2*n - 2)*x^4)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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_A013609 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 A013609 → 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.