A013612 → x ↦ x+1 → basis A013610

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:          121,           330,            225;
  1:         1331,          5445,           7425,            3375;
  2:        14641,         79860,         163350,          148500,           50625;
  3:       161051,       1098075,        2994750,         4083750,         2784375,          759375;
  4:      1771561,      14494590,       49413375,        89842500,        91884375,        50118750,        11390625;
  5:     19487171,     186013905,      760965975,      1729468125,      2358365625,      1929571875,       877078125,       170859375;
  6:    214358881,    2338460520,    11160834300,     30438639000,     51884043750,     56600775000,     38591437500,     15035625000,      2562890625;
  7:   2357947691,   28938448935,   157846085100,    502237543500,   1027304066250,   1400869181250,   1273517437500,    744263437500,    253726171875,     38443359375;
  8:  25937424601,  353692153650,  2170383670125,   7892304255000,  18833907881250,  30819121987500,  35021729531250,  27289659375000,  13954939453125,   4228769531250,   576650390625;
  9: 285311670611, 4279675059165, 29179602676125, 119371101856875, 325557550518750, 621518960081250, 847525854656250, 825512196093750, 562849224609375, 255840556640625, 69774697265625, 8649755859375;

The row polynomials satisfy P(n,x) = (11+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 330*x + 225*x^2.

P[0, x_] := 121 + 330*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 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] = 121 + 330*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 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 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_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
        • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038259 → basis A013610
• A038259 → basis A013610 → D
• A038271 → basis A013609 → basis A013610
• A038259 → basis A013610 → reverse
• A013618 → basis A013610 → x ↦ x+1
• A038336 → basis A013610

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.