A159854 → x ↦ x+1 → (θ+1)²

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:   1,     8,     9;
  1:   2,    20,    36,     16;
  2:   4,    48,   117,     96,     25;
  3:   8,   112,   342,    400,    200,     36;
  4:  16,   256,   936,   1408,   1025,    360,     49;
  5:  32,   576,  2448,   4480,   4250,   2196,    588,     64;
  6:  64,  1280,  6192,  13312,  15500,  10512,   4165,    896,    81;
  7: 128,  2816, 15264,  37632,  51800,  43344,  22638,   7232,  1296,   100;
  8: 256,  6144, 36864, 102400, 162400, 161280, 104272,  44032, 11745,  1800,  121;
  9: 512, 13312, 87552, 270336, 484800, 556416, 428064, 224256, 79218, 18100, 2420, 144;

The row polynomials satisfy (4+(6*n+30)*x+(n^2+20*n+48)*x^2+(2*n^2+18*n+24)*x^3+(n^2+4*n+4)*x^4)*P(n,x) = (8+(12*n+76)*x+(2*n^2+50*n+174)*x^2+(5*n^2+66*n+157)*x^3+(4*n^2+34*n+62)*x^4+(n^2+6*n+9)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 9*x^2.

P[0, x_] := 1 + 8*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8 + (12*n + 76)*x + (2*n^2 + 50*n + 174)*x^2 + (5*n^2 + 66*n + 157)*x^3 + (4*n^2 + 34*n + 62)*x^4 + (n^2 + 6*n + 9)*x^5)*P[n - 1, x])/(4 + (6*n + 30)*x + (n^2 + 20*n + 48)*x^2 + (2*n^2 + 18*n + 24)*x^3 + (n^2 + 4*n + 4)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 8*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (12*n + 76)*x + (2*n^2 + 50*n + 174)*x^2 + (5*n^2 + 66*n + 157)*x^3 + (4*n^2 + 34*n + 62)*x^4 + (n^2 + 6*n + 9)*x^5)*v[i - 1])/(4 + (6*n + 30)*x + (n^2 + 20*n + 48)*x^2 + (2*n^2 + 18*n + 24)*x^3 + (n^2 + 4*n + 4)*x^4)); 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 shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • Parent-row provers of polytool polytool ? step › direct check
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

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.