A038305 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
  0:           247,             96,              9;
  1:          3718,           2223,            432,             27;
  2:         54925,          44616,          13338,           1728,            81;
  3:        799708,         823875,         334620,          66690,          6480,           243;
  4:      11510083,       14394744,        7414875,        2007720,        300105,         23328,          729;
  5:     164111506,      241711743,      151144812,       51904125,      10540530,       1260441,        81648,        2187;
  6:    2321695129,     3938676144,     2900540916,     1209158496,     311424750,      50594544,      5041764,      279936,       6561;
  7:   32629228840,    62685768483,    53172127944,    26104868244,    8161819848,    1681693650,    227675448,    19446804,     944784,     19683;
  8:  455993473039,   978876865200,   940286527245,   531721279440,  195786511830,   48970919088,   8408468250,   975751920,   72925515,   3149280,    59049;
  9: 6341490625054, 15047784610287, 16151468275800, 10343151799695, 4386700555380, 1292190978078, 269340054984, 39639921750, 4024976670, 267393555, 10392624, 177147;

The row polynomials satisfy (3*n+16+3*x)*P(n,x) = (39*n+247+(9*n+96)*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 247 + 96*x + 9*x^2.

P[0, x_] := 247 + 96*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((39*n + 247 + (9*n + 96)*x + 9*x^2)*P[n - 1, x])/(3*n + 16 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 247 + 96*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((39*n + 247 + (9*n + 96)*x + 9*x^2)*v[i - 1])/(3*n + 16 + 3*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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
        • 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 Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038327 → basis A013610 → D+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.