A013612 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
  0:        36,         360,          675;
  1:       216,        3240,        12150,         13500;
  2:      1296,       25920,       145800,        324000,         253125;
  3:      7776,      194400,      1458000,       4860000,        7593750,        4556250;
  4:     46656,     1399680,     13122000,      58320000,      136687500,      164025000,        79734375;
  5:    279936,     9797760,    110224800,     612360000,     1913625000,     3444525000,      3348843750,      1366875000;
  6:   1679616,    67184640,    881798400,    5878656000,    22963500000,    55112400000,     80372250000,     65610000000,     23066015625;
  7:  10077696,   453496320,   6802444800,   52907904000,   248005800000,   744017400000,   1446700500000,   1771470000000,   1245564843750,    384433593750;
  8:  60466176,  3023308800,  51018336000,  453496320000,  2480058000000,  8928208800000,  21700507500000,  35429400000000,  37366945312500,  23066015625000,   6343154296875;
  9: 362797056, 19953838080, 374134464000, 3741344640000, 23383404000000, 98210296800000, 286446699000000, 584585100000000, 822072796875000, 761178515625000, 418648183593750, 103797070312500;

The row polynomials satisfy (2+(5*n+10)*x)*P(n,x) = (12+(30*n+120)*x+(75*n+225)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 360*x + 675*x^2.

P[0, x_] := 36 + 360*x + 675*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + (30*n + 120)*x + (75*n + 225)*x^2)*P[n - 1, x])/(2 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 360*x + 675*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + (30*n + 120)*x + (75*n + 225)*x^2)*v[i - 1])/(2 + (5*n + 10)*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
• 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_A013610 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
            • 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
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

• A013622 → basis A013611 → 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. [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.