A038319 → θ+1

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Submission text ?

Triangle begins:
  0:          121,           220,            75;
  1:         1331,          3630,          2475,            500;
  2:        14641,         53240,         54450,          22000,           3125;
  3:       161051,        732050,        998250,         605000,         171875,          18750;
  4:      1771561,       9663060,      16471125,       13310000,        5671875,        1237500,        109375;
  5:     19487171,     124009270,     253655325,      256217500,      145578125,       47643750,       8421875,        625000;
  6:    214358881,    1558973680,    3720278100,     4509428000,     3202718750,     1397550000,     370562500,      55000000,      3515625;
  7:   2357947691,   19292299290,   52615361700,    74405562000,    63413831250,    34589362500,   12228562500,    2722500000,    348046875,     19531250;
  8:  25937424601,  235794769100,  723461223375,  1169230260000,  1162586906250,   760965975000,  336285468750,   99825000000,  19142578125,   2148437500,   107421875;
  9: 285311670611, 2853116706110, 9726534225375, 17684607682500, 20096145093750, 15346147162500, 8138108343750, 3019706250000, 772083984375, 129980468750, 12998046875, 585937500;

The row polynomials satisfy (11+(5*n+10)*x)*P(n,x) = (121+(55*n+220)*x+(25*n+75)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 220*x + 75*x^2.

P[0, x_] := 121 + 220*x + 75*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((121 + (55*n + 220)*x + (25*n + 75)*x^2)*P[n - 1, x])/(11 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 121 + 220*x + 75*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (55*n + 220)*x + (25*n + 75)*x^2)*v[i - 1])/(11 + (5*n + 10)*x)); 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 Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 ?

• A038259 → basis A007318 → θ+1
• A038336 → x ↦ x+1 → θ+1

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.