A038329 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:      9,      90,       225;
  1:     27,     405,      2025,       3375;
  2:     81,    1620,     12150,      40500,       50625;
  3:    243,    6075,     60750,     303750,      759375,       759375;
  4:    729,   21870,    273375,    1822500,     6834375,     13668750,      11390625;
  5:   2187,   76545,   1148175,    9568125,    47840625,    143521875,     239203125,     170859375;
  6:   6561,  262440,   4592700,   45927000,   287043750,   1148175000,    2870437500,    4100625000,     2562890625;
  7:  19683,  885735,  17714700,  206671500,  1550036250,   7750181250,   25833937500,   55358437500,    69198046875,    38443359375;
  8:  59049, 2952450,  66430125,  885735000,  7750181250,  46501087500,  193754531250,  553584375000,  1037970703125,  1153300781250,   576650390625;
  9: 177147, 9743085, 243577125, 3653656875, 36536568750, 255755981250, 1278779906250, 4567071093750, 11417677734375, 19029462890625, 19029462890625, 8649755859375;

The row polynomials satisfy P(n,x) = (3+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 90*x + 225*x^2.

P[0, x_] := 9 + 90*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 9 + 90*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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_A007318 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A013612 → D
• A038291 → basis A013609
• basis-map coefficients of A112227 → basis A013611 → x ↦ x/(x+1)
• A027467 → basis A013611 → x ↦ x+1
• A130123 → basis A013611 → x ↦ x/(x+1)
• A013612 → reverse
• A136572 → basis A013611 → x ↦ x/(x+1)
• A038243 → reverse
• A038291 → basis A013609 → D
• A147716 → basis A013610 → D
• A038329 → x ↦ x+1
• Triangle of coefficients in expansion of (1+5x)^n.
• A147716 → basis A013610 → reverse
• A038330 → basis A007318 → basis A007318
• A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A013611 → x ↦ x/(x+1)
• A038291 → basis A013609 → reverse
• A038243 → D
• A147716 → basis A013610
• A038329 → x ↦ x+1 → D
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).

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.