A134346 → basis A008277

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No OEIS matches.

Submission text ?

Triangle begins:
  0:      1;
  1:      6,        3;
  2:     28,       35,         7;
  3:    120,      285,       135,        15;
  4:    496,     2015,      1705,       434,        31;
  5:   2016,    13293,     17955,      7875,      1260,       63;
  6:   8128,    84455,    171577,    115570,     31115,     3429,      127;
  7:  32640,   525045,   1547595,   1504755,    606900,   110670,     8925,     255;
  8: 130816,  3221855,  13457185,  18178314,  10312491,  2768598,   364854,   22484,   511;
  9: 523776, 19611933, 114212835, 208901715, 160263180, 59443461, 11385990, 1135530, 55242, 1023;

The row polynomials satisfy P(n,x) = -(8+10*x+2*x^2)*P(n-2,x) - (10*x+4*x^2)*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.

P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 10*x + 2*x^2)*P[n - 2, x] - (10*x + 4*x^2)*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 10*x + 2*x^2)*v[i - 2] - (10*x + 4*x^2)*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (6 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
    • 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 Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
        • 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 Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A376827 → basis A008277
• A028326 → basis A013609 → basis A008277
• A038208 → basis A008277
• A178821 → basis A008277
• A134059 → basis A008277
• A003506 → D → basis A008277
• A178821 → basis A013609 → basis A008277
• A028326 → basis A008277
• A178822 → basis A013609 → basis A008277
• A003506 → basis A008277
• A132200 → basis A008277
• A038338 → basis A013609 → basis A008277
• basis-map coefficients of A127647 → basis A008277
• A038221 → basis A013609 → basis A008277
• A038338 → basis A008277
• basis-map coefficients of A113214, shifted → D → basis A008277
• A027465 → basis A013611 → basis A008277
• A196347 → basis A008277
• A258758 → basis A008277
• A196347 → basis A013609 → basis A008277
• A156992 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A135838 → basis A008277
• A094305 → basis A008277
• A038208 → D → basis A008277
• A028326 → D → basis A008277
• A085880 → basis A013609 → basis A008277
• A124860 → basis A008277
• A178822 → basis A008277
• A135065 → basis A008277
• A126351 → reverse
• A135089 → basis A008277
• A134058 → basis A008277
• A038330 → basis A013611 → basis A008277
• A178820 → basis A013609 → basis A008277
• A038325 → basis A008277
• A085881 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A038325 → basis A013609 → basis A008277
• A143494 → reverse
• A344565 → basis A008277
• A356546 → basis A013609 → basis A008277
• A178820 → basis A008277
• A085881 → basis A013609 → basis A008277
• basis-map coefficients of A021012 → basis A008277 → D+1
• A085880 → basis A008277
• A120910 → basis A013610 → basis A008277
• basis-map coefficients of A113214, shifted → basis A013609 → basis A008277
• basis-map coefficients of A026729, shifted → basis A007318 → basis A008277
• A131431 → x ↦ x+1 → basis A008277
• A356546 → basis A008277
• basis-map coefficients of A112227 → x ↦ x+1 → basis A008277
• A038221 → basis A008277
• A085880 → D → basis A008277
• A156992 → basis A013609 → basis A008277
• A156992 → basis A008277 → reverse

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741