A069466 → γ → (θ+1)²

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Submission text ?

Triangle begins:
  0:     1;
  1:     2;
  2:     6,       48;
  3:    20,      480;
  4:    70,     3360,      3780;
  5:   252,    20160,     68040;
  6:   924,   110880,    748440,     295680;
  7:  3432,   576576,   6486480,    7687680;
  8: 12870,  2882880,  48648600,  115315200,  22522500;
  9: 48620, 14002560, 330810480, 1306905600, 765765000;

The row polynomials satisfy (n^2-(4*n^4-2*n^2)*x+(n^6+2*n^5+n^4)*x^2)*P(n,x) = (4*n^2-2*n-(16*n^4-8*n^3-72*n^2+100*n-32)*x+(4*n^6+6*n^5-112*n^4-42*n^3+224*n^2-56*n-16)*x^2+(16*n^6+72*n^5+72*n^4-72*n^3-120*n^2+32)*x^3)*P(n-1,x) + ((8*n-4)*x-(32*n^3+48*n^2+16*n-24)*x^2+(8*n^5+44*n^4+208*n^3+236*n^2-80*n-48)*x^3-(32*n^5+176*n^4+320*n^3+176*n^2-64*n-64)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n - (16*n^4 - 8*n^3 - 72*n^2 + 100*n - 32)*x + (4*n^6 + 6*n^5 - 112*n^4 - 42*n^3 + 224*n^2 - 56*n - 16)*x^2 + (16*n^6 + 72*n^5 + 72*n^4 - 72*n^3 - 120*n^2 + 32)*x^3)*P[n - 1, x] + ((8*n - 4)*x - (32*n^3 + 48*n^2 + 16*n - 24)*x^2 + (8*n^5 + 44*n^4 + 208*n^3 + 236*n^2 - 80*n - 48)*x^3 - (32*n^5 + 176*n^4 + 320*n^3 + 176*n^2 - 64*n - 64)*x^4)*D[P[n - 1, x], x])/(n^2 - (4*n^4 - 2*n^2)*x + (n^6 + 2*n^5 + n^4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n - (16*n^4 - 8*n^3 - 72*n^2 + 100*n - 32)*x + (4*n^6 + 6*n^5 - 112*n^4 - 42*n^3 + 224*n^2 - 56*n - 16)*x^2 + (16*n^6 + 72*n^5 + 72*n^4 - 72*n^3 - 120*n^2 + 32)*x^3)*v[i - 1] + ((8*n - 4)*x - (32*n^3 + 48*n^2 + 16*n - 24)*x^2 + (8*n^5 + 44*n^4 + 208*n^3 + 236*n^2 - 80*n - 48)*x^3 - (32*n^5 + 176*n^4 + 320*n^3 + 176*n^2 - 64*n - 64)*x^4)*deriv(v[i - 1], x))/(n^2 - (4*n^4 - 2*n^2)*x + (n^6 + 2*n^5 + n^4)*x^2)); 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

Proof chains ? JSON

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

  • from gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps an…?
    • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

      • from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Three-term recurrence criterion (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A089627 → (θ+1)²
• A391403 → (θ+1)²

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