A102413 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:  0;
  1:  1;
  2:  4,   2;
  3:  6,  12,    3;
  4:  8,  32,   24,    4;
  5: 10,  60,   90,   40,    5;
  6: 12,  96,  228,  192,   60,    6;
  7: 14, 140,  462,  616,  350,   84,   7;
  8: 16, 192,  816, 1536, 1360,  576, 112,   8;
  9: 18, 252, 1314, 3240, 4050, 2628, 882, 144, 9;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + (3*n^2 - 10*n)*x - 2*n*x^2)*v[i - 1] + (n*x + 6*n*x^2 + n*x^3)*deriv(v[i - 1], x))/(-n^2 + 3*n - 2 + (n^2 - n)*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 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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)

Related ?

Same fingerprint ?

• A113413 → reverse
• A122542 → reverse
• A035607 → reverse
• A266213: Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, …
• A035607: Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
• A113413: A Riordan array of coordination sequences.
• A122542: Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0…

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.