A104978 → θ+1

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Triangle begins:
  0:    1;
  1:    1,      2;
  2:    2,     10,       9;
  3:    5,     42,      84,      48;
  4:   14,    168,     540,     660,      275;
  5:   42,    660,    2970,    5720,     5005,     1638;
  6:  132,   2574,   15015,   40040,    54600,    37128,     9996;
  7:  429,  10010,   72072,  247520,   464100,   488376,   271320,    62016;
  8: 1430,  38896,  334152, 1410864,  3391500,  4883760,  4178328,  1961256,   389367;
  9: 4862, 151164, 1511640, 7596960, 22383900, 41186376, 48050772, 34610400, 14060475, 2466750;

The row polynomials satisfy (4*n^2+2*n+(8*n^3+4*n^2)*x)*P(n,x) = (12*n^2-6*n+(28*n^3+18*n^2-28*n+6)*x+(36*n^3+6*n^2-24*n+6)*x^2)*P(n-1,x) + (n+(2*n^2+8*n-3)*x+(20*n^2+19*n-9)*x^2+(18*n^2+12*n-6)*x^3)*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[((12*n^2 - 6*n + (28*n^3 + 18*n^2 - 28*n + 6)*x + (36*n^3 + 6*n^2 - 24*n + 6)*x^2)*P[n - 1, x] + (n + (2*n^2 + 8*n - 3)*x + (20*n^2 + 19*n - 9)*x^2 + (18*n^2 + 12*n - 6)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (8*n^3 + 4*n^2)*x)]];
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] = ((12*n^2 - 6*n + (28*n^3 + 18*n^2 - 28*n + 6)*x + (36*n^3 + 6*n^2 - 24*n + 6)*x^2)*v[i - 1] + (n + (2*n^2 + 8*n - 3)*x + (20*n^2 + 19*n - 9)*x^2 + (18*n^2 + 12*n - 6)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (8*n^3 + 4*n^2)*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

Proof chains ? JSON

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

  • from Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
        • 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 read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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
    • ↑ Iterated real-rootedness preservers (above)
    • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A330798 → θ+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.