A038329 → basis A013610 → reverse

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Submission text ?

Triangle begins:
  0:          81,          270,           225;
  1:         729,         3645,          6075,           3375;
  2:        6561,        43740,        109350,         121500,          50625;
  3:       59049,       492075,       1640250,        2733750,        2278125,          759375;
  4:      531441,      5314410,      22143375,       49207500,       61509375,        41006250,        11390625;
  5:     4782969,     55801305,     279006525,      775018125,     1291696875,      1291696875,       717609375,       170859375;
  6:    43046721,    573956280,    3348078300,    11160261000,    23250543750,     31000725000,     25833937500,     12301875000,      2562890625;
  7:   387420489,   5811307335,   38742048900,   150663523500,   376658808750,    627764681250,    697516312500,    498225937500,    207594140625,     38443359375;
  8:  3486784401,  58113073350,  435848050125,  1937102445000,  5649882131250,  11299764262500,  15694117031250,  14946778125000,   9341736328125,   3459902343750,   576650390625;
  9: 31381059609, 575319426165, 4794328551375, 23971642756875, 79905475856250, 186446110331250, 310743517218750, 369932758593750, 308277298828125, 171265166015625, 57088388671875, 8649755859375;

The row polynomials satisfy P(n,x) = (9+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 81 + 270*x + 225*x^2.

P[0, x_] := 81 + 270*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(9 + 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] = 81 + 270*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (9 + 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 (−∞, 0]
  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 Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • 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 ?

• A038334 → x ↦ x/(x+1)
• A196347 → basis A013610 → basis A007318
• A013612 → basis A013609 → D
• A120910 → basis A013609 → x ↦ x/(x+1)
• A013612 → basis A013609 → reverse
• A038271 → basis A013611
• A038271 → basis A013610 → basis A013610
• A038329 → basis A013610 → D
• A038338 → basis A013610 → x ↦ x+1
• A156992 → basis A013610 → x ↦ x+1
• A178822 → basis A013610 → basis A007318
• A013620 → basis A007318
• A038231 → D → basis A013610
• A038243 → basis A013611 → x ↦ x/(x+1)
• A027467 → basis A013611 → basis A013610
• A178820 → basis A013610 → x ↦ x+1
• A038335 → x ↦ x+1 → basis A013609
• A038271 → basis A013611 → D
• A085880 → basis A013610 → x ↦ x+1
• A013620 → basis A007318 → D
• A038319 → basis A013611 → basis A013610
• A013622 → basis A013609 → basis A013610
• A038284 → basis A007318 → basis A013609
• A178821 → basis A013610 → x ↦ x+1
• A028326 → basis A013610 → x ↦ x+1
• A038221 → basis A013610 → basis A007318
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
• A013612 → basis A013609
• A038329 → basis A013610
• A013622 → D
• A038255 → basis A013610 → basis A013609

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.