A038321 → reverse

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Submission text ?

Triangle begins:
  0:         49,         154,          121;
  1:        343,        1617,         2541,          1331;
  2:       2401,       15092,        35574,         37268,         14641;
  3:      16807,      132055,       415030,        652190,        512435,        161051;
  4:     117649,     1109262,      4357815,       9130660,      10761135,       6764142,        1771561;
  5:     823543,     9058973,     42706587,     111850585,     175765205,     165721479,       86806489,       19487171;
  6:    5764801,    72471784,    398594812,    1252726552,    2460712870,    3093467608,     2430581692,     1091281576,      214358881;
  7:   40353607,   570715299,   3587353308,   13153628796,   31004982162,   48722114826,    51042215532,    34375369644,    13504609503,    2357947691;
  8:  282475249,  4438896770,  31389341445,  131536287960,  361724791890,  682109607564,   893238771810,   802091958360,   472661332605,  165056338370,   25937424601;
  9: 1977326743, 34179505129, 268553254585, 1266036771615, 3978972710790, 8753739963738, 13755877085874, 15440270198430, 12131640870195, 6354669027245, 1997181694277, 285311670611;

The row polynomials satisfy P(n,x) = (7+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 49 + 154*x + 121*x^2.

P[0, x_] := 49 + 154*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(7 + 11*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 49 + 154*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + 11*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 Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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
    • 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

Related ?

Same fingerprint ?

• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
• A038328 → basis A013611 → x ↦ x/(x+1)
• A038321 → D
• A038301 → basis A013609 → basis A013610
• A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
• A038277 → D
• A038277 → reverse

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.