A038319 → basis A013610 → basis A013610

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Triangle begins:
  0:               961,               2790,                2025;
  1:             29791,             129735,              188325,                91125;
  2:            923521,            5362380,            11676150,             11299500,              4100625;
  3:          28629151,          207792225,           603267750,            875711250,            635596875,            184528125;
  4:         887503681,         7729870770,         28051950375,          54294097500,          59110509375,          34322231250,            8303765625;
  5:       27512614111,       279563659515,       1217454646275,        2945454789375,        4275660178125,        3723962090625,         1801917140625,          373669453125;
  6:      852891037441,      9904541079960,      50321458712700,      146094557553000,      265090931043750,      307847532825000,       223437725437500,        92670024375000,       16815125390625;
  7:    26439622160671,    345420870163605,    2005669568691900,     6793396926214500,    14792073952241250,    21472365414543750,     20779708465687500,     12927468400312500,     4691419983984375,      756680642578125;
  8:   819628286980801,  11897829972301950,   77719695786811125,   300850435303785000,   764257154199131250,  1331286655701712500,   1610427406090781250,   1335838401365625000,   727170097517578125,   234570999199218750,    34050628916015625;
  9: 25408476896404831, 405716002055496495, 2944712918144732625, 12823749804823835625, 37230241368843393750, 75661458265713993750, 109831149095391281250, 113880223716419531250, 82655001084498046875, 39994355363466796875, 11611264460361328125, 1532278301220703125;

The row polynomials satisfy P(n,x) = (31+45*x)*P(n-1,x) for n >= 1, with P(0,x) = 961 + 2790*x + 2025*x^2.

P[0, x_] := 961 + 2790*x + 2025*x^2;
P[n_, x_] := P[n, x] = Expand[(31 + 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] = 961 + 2790*x + 2025*x^2; for(n = 1, N, my(i = n + 1); v[i] = (31 + 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)*11^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(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
        • 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)
        • 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 basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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.