A013618 → reverse

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Submission text ?

Triangle begins:
  0:          121,           22,            1;
  1:         1331,          363,           33,           1;
  2:        14641,         5324,          726,          44,          1;
  3:       161051,        73205,        13310,        1210,         55,         1;
  4:      1771561,       966306,       219615,       26620,       1815,        66,        1;
  5:     19487171,     12400927,      3382071,      512435,      46585,      2541,       77,       1;
  6:    214358881,    155897368,     49603708,     9018856,    1024870,     74536,     3388,      88,      1;
  7:   2357947691,   1929229929,    701538156,   148811124,   20292426,   1844766,   111804,    4356,     99,    1;
  8:  25937424601,  23579476910,   9646149645,  2338460520,  372027810,  40584852,  3074610,  159720,   5445,  110,   1;
  9: 285311670611, 285311670611, 129687123005, 35369215365, 6430766430, 818461182, 74405562, 4831530, 219615, 6655, 121, 1;

The row polynomials satisfy P(n,x) = (11+x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 22*x + x^2.

P[0, x_] := 121 + 22*x + x^2;
P[n_, x_] := P[n, x] = Expand[(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] = 121 + 22*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (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 of coefficients in expansion of (1+11x)^n.?
    • 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 ?

• A013618 → D
• A038303 → basis A007318 → reverse
• Triangle of coefficients in expansion of (1+11x)^n.
• A038303 → basis A007318
• A038303 → basis A007318 → D

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.