A038271 → basis A013610 → basis A013610

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Submission text ?

Triangle begins:
  0:             1;
  1:            27,              45;
  2:           729,            2430,            2025;
  3:         19683,           98415,          164025,            91125;
  4:        531441,         3542940,         8857350,          9841500,          4100625;
  5:      14348907,       119574225,       398580750,        664301250,        553584375,         184528125;
  6:     387420489,      3874204890,     16142520375,      35872267500,      44840334375,       29893556250,        8303765625;
  7:   10460353203,    122037454035,    610187270175,    1694964639375,    2824941065625,     2824941065625,     1569411703125,     373669453125;
  8:  282429536481,   3765727153080,  21966741726300,   73222472421000,  152546817543750,   203395756725000,   169496463937500,   80712601875000,   16815125390625;
  9: 7625597484987, 114383962274805, 762559748498700, 2965510133050500, 7413775332626250, 12356292221043750, 13729213578937500, 9806581127812500, 4086075469921875, 756680642578125;

The row polynomials satisfy P(n,x) = (27+45*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 + 45*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 + 45*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from 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

Related ?

Same fingerprint ?

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