A102413 → basis A007318

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:    1;
  1:    2,     1;
  2:    6,     6,     1;
  3:   14,    21,     9,     1;
  4:   34,    68,    46,    12,     1;
  5:   82,   205,   190,    80,    15,    1;
  6:  198,   594,   699,   408,   123,   18,    1;
  7:  478,  1673,  2387,  1785,   749,  175,   21,   1;
  8: 1154,  4616,  7740,  7064,  3814, 1240,  236,  24,  1;
  9: 2786, 12537, 24156, 26040, 17226, 7218, 1908, 306, 27, 1;

The row polynomials satisfy (n-1)*P(n,x) = (2*n^2-5*n+3)*P(n-1,x) - (8*n-20+(8*n-21)*x+(n-3)*x^2)*d/dx P(n-1,x) + (8+16*x+9*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3)*P[n - 1, x] - (8*n - 20 + (8*n - 21)*x + (n - 3)*x^2)*D[P[n - 1, x], x] + (8 + 16*x + 9*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 3)*v[i - 1] - (8*n - 20 + (8*n - 21)*x + (n - 3)*x^2)*deriv(v[i - 1], x) + (8 + 16*x + 9*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
• Ore-algebra identity of two definitions polytool ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
    • ↑ Liu–Wang theorem for three-term recurrences (above)
    • ↑ Liu–Wang theorem with a lag row (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A102413 → x ↦ x/(x+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