A038319 → reverse

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Submission text ?

Triangle begins:
  0:       25,        110,         121;
  1:      125,        825,        1815,        1331;
  2:      625,       5500,       18150,       26620,        14641;
  3:     3125,      34375,      151250,      332750,       366025,        161051;
  4:    15625,     206250,     1134375,     3327500,      5490375,       4831530,       1771561;
  5:    78125,    1203125,     7940625,    29115625,     64054375,      84551775,      62004635,      19487171;
  6:   390625,    6875000,    52937500,   232925000,    640543750,    1127357000,    1240092700,     779486840,     214358881;
  7:  1953125,   38671875,   340312500,  1746937500,   5764893750,   12682766250,   18601390500,   17538453900,    9646149645,    2357947691;
  8:  9765625,  214843750,  2126953125, 12478125000,  48040781250,  126827662500,  232517381250,  292307565000,  241153741125,  117897384550,   25937424601;
  9: 48828125, 1181640625, 12998046875, 85787109375, 377463281250, 1162586906250, 2557691193750, 4019229018750, 4421151920625, 3242178075125, 1426558353055, 285311670611;

The row polynomials satisfy P(n,x) = (5+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 110*x + 121*x^2.

P[0, x_] := 25 + 110*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 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] = 25 + 110*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 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)*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)

Related ?

Same fingerprint ?

• A038336 → x ↦ x+1 → D
• A038301 → basis A013611
• A013618 → basis A013610 → basis A013610
• A038336 → x ↦ x+1 → reverse
• A038336 → x ↦ x+1
• A038259 → basis A007318
• A038301 → basis A013611 → D
• A038301 → basis A013611 → reverse
• A038271 → basis A013609 → x ↦ x+1
• A038220 → basis A013610 → x ↦ x/(x+1)
• A038319 → D
• A038314 → x ↦ x/(x+1)
• A038259 → basis A007318 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• A038259 → basis A007318 → 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.