A133214 → D

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Submission text ?

Triangle begins:
  0:   8,     8;
  1:  18,    72,     24;
  2:  32,   288,    384,      64;
  3:  50,   800,   2400,    1600,      160;
  4:  72,  1800,   9600,   14400,     5760,      384;
  5:  98,  3528,  29400,   78400,    70560,    18816,      896;
  6: 128,  6272,  75264,  313600,   501760,   301056,    57344,     2048;
  7: 162, 10368, 169344, 1016064,  2540160,  2709504,  1161216,   165888,     4608;
  8: 200, 16200, 345600, 2822400, 10160640, 16934400, 12902400,  4147200,   460800,   10240;
  9: 242, 24200, 653400, 6969600, 34151040, 81962496, 97574400, 55756800, 13939200, 1239040, 22528;

The row polynomials satisfy (n^2+2*n+1)*P(n,x) = (n^2+4*n+4+(6*n^2+12*n+4)*x)*P(n-1,x) + ((2*n+3)*x-(4*n+6)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8 + 8*x.

P[0, x_] := 8 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 4 + (6*n^2 + 12*n + 4)*x)*P[n - 1, x] + ((2*n + 3)*x - (4*n + 6)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 8 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 4 + (6*n^2 + 12*n + 4)*x)*v[i - 1] + ((2*n + 3)*x - (4*n + 6)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*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 (−∞, 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 ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from Delannoy paths counted by number of weak peaks.?
    • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? 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
    • 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

Related ?

Same fingerprint ?

• A114655 → θ+1
• A114655 → θ+1 → reverse
• A114687 → reverse → θ+1
• A114656 → θ+1 → reverse
• A114656 → 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. [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