row-factor cofactor of A101842 → θ+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0: 1;
  1: 1,    2;
  2: 1,    8,      3;
  3: 1,   22,     33,       4;
  4: 1,   52,    198,     104,       5;
  5: 1,  114,    906,    1208,     285,       6;
  6: 1,  240,   3573,    9664,    5955,     720,       7;
  7: 1,  494,  12879,   62476,   78095,   25758,    1729,      8;
  8: 1, 1004,  43824,  352936,  780950,  529404,  102256,   4016,    9;
  9: 1, 2026, 143520, 1820768, 6551770, 7862124, 3186344, 382720, 9117, 10;

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

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 2, x] - ((n + 1)*x^2 - (n + 1)*x^3)*D[P[n - 2, x], x] + (1 + 2*n*x)*P[n - 1, x] + (x - x^2)*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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 2] - ((n + 1)*x^2 - (n + 1)*x^3)*deriv(v[i - 2], x) + (1 + 2*n*x)*v[i - 1] + (x - x^2)*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 Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Parent-row provers of polytool polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A008292 → θ+1 (shift +2)
• A173018 → θ+1
• A008292 → θ+1 (shift +2) → reverse
• A008292 → θ+1
• A257620 → θ+1
• A257609 → θ+1
• A334218 → D
• A173018 → reverse (shift +1) → reverse → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.