A013612 → basis A013609 → (θ+1)²

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Submission text ?

Triangle begins:
  0:        36,         480,          900;
  1:       216,        4320,        16200,         16000;
  2:      1296,       34560,       194400,        384000,         250000;
  3:      7776,      259200,      1944000,       5760000,        7500000,        3600000;
  4:     46656,     1866240,     17496000,      69120000,      135000000,      129600000,        49000000;
  5:    279936,    13063680,    146966400,     725760000,     1890000000,     2721600000,      2058000000,       640000000;
  6:   1679616,    89579520,   1175731200,    6967296000,    22680000000,    43545600000,     49392000000,     30720000000,      8100000000;
  7:  10077696,   604661760,   9069926400,   62705664000,   244944000000,   587865600000,    889056000000,    829440000000,    437400000000,    100000000000;
  8:  60466176,  4031078400,  68024448000,  537477120000,  2449440000000,  7054387200000,  13335840000000,  16588800000000,  13122000000000,   6000000000000,  1210000000000;
  9: 362797056, 26605117440, 498845952000, 4434186240000, 23094720000000, 77598259200000, 176033088000000, 273715200000000, 288684000000000, 198000000000000, 79860000000000, 14400000000000;

The row polynomials satisfy (9+(45*n+75)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (54+(270*n+810)*x+(150*n^2+1350*n+2550)*x^2+(250*n^2+1500*n+2250)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 480*x + 900*x^2.

P[0, x_] := 36 + 480*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((54 + (270*n + 810)*x + (150*n^2 + 1350*n + 2550)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*P[n - 1, x])/(9 + (45*n + 75)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 480*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((54 + (270*n + 810)*x + (150*n^2 + 1350*n + 2550)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*v[i - 1])/(9 + (45*n + 75)*x + (25*n^2 + 100*n + 100)*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
• 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 basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
        • 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

Related ?

Same fingerprint ?

• A038271 → basis A013611 → (θ+1)²
• A013622 → (θ+1)²

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. [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.