A038271 → basis A013610 → basis A007318

Matches ?

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Submission text ?

Triangle begins:
  0:             1;
  1:            27,             15;
  2:           729,            810,            225;
  3:         19683,          32805,          18225,            3375;
  4:        531441,        1180980,         984150,          364500,          50625;
  5:      14348907,       39858075,       44286750,        24603750,        6834375,         759375;
  6:     387420489,     1291401630,     1793613375,      1328602500,      553584375,      123018750,       11390625;
  7:   10460353203,    40679151345,    67798585575,     62776468125,    34875815625,    11625271875,     2152828125,     170859375;
  8:  282429536481,  1255242384360,  2440749080700,   2711943423000,  1883294043750,   837019575000,   232505437500,   36905625000,   2562890625;
  9: 7625597484987, 38127987424935, 84728860944300, 109833708631500, 91528090526250, 50848939181250, 18832940437500, 4484033437500, 622782421875, 38443359375;

The row polynomials satisfy P(n,x) = (27+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(27 + 15*x)*P[n - 1, 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] = (27 + 15*x)*v[i - 1]); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A013620 → basis A013610 → D
• A196347 → basis A013610 → basis A013610
• A038338 → basis A013610 → basis A013610
• A038334 → basis A013609 → x ↦ x/(x+1)
• A178822 → basis A013610 → basis A013610
• A038319 → basis A013611 → x ↦ x+1
• A028326 → basis A013610 → basis A013610
• A038323 → basis A013611
• A038329 → basis A013611 → x ↦ x/(x+1)
• A013620 → basis A013610
• A013620 → basis A013610 → reverse
• A038221 → basis A013610 → basis A013610
• A038208 → basis A013610 → basis A013610
• A013622 → basis A013609 → basis A007318
• A178820 → basis A013610 → basis A013610
• A085880 → basis A013610 → basis A013610
• A178821 → basis A013610 → basis A013610
• A156992 → basis A013610 → basis A013610

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.