A114687 → basis A013611 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:        0;
  1:        8;
  2:       56,       128;
  3:      336,      1536,       1536;
  4:     1936,     13312,      27648,      16384;
  5:    11000,    101120,     322560,     409600,      163840;
  6:    62184,    716160,    3095040,    6144000,     5406720,     1572864;
  7:   351008,   4859904,   26557440,   72253440,   100925440,    66060288,    14680064;
  8:  1981472,  32055296,  212029440,  735379456,  1422131200,  1497366528,   763363328,  134217728;
  9: 11194344, 207226368, 1609961472, 6800080896, 16886661120, 24838668288, 20610809856, 8455716864, 1207959552;

The row polynomials satisfy (4*n^2+4*n-8)*P(n,x) = (28*n^2+4*n-8+(96*n^2-32*n-64)*x)*P(n-1,x) - (2*n+1+(24*n+12)*x+(64*n+32)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.

P[0, x_] := 0;
P[1, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((28*n^2 + 4*n - 8 + (96*n^2 - 32*n - 64)*x)*P[n - 1, x] - (2*n + 1 + (24*n + 12)*x + (64*n + 32)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 4*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((28*n^2 + 4*n - 8 + (96*n^2 - 32*n - 64)*x)*v[i - 1] - (2*n + 1 + (24*n + 12)*x + (64*n + 32)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 4*n - 8)); 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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013611 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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 reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two c…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023