A038305 → basis A013609 → D+1

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Submission text ?

Triangle begins:
  0:            325,            228,              36;
  1:           5239,           5850,            2052,             216;
  2:          81289,         125736,           70200,           16416,            1296;
  3:        1228123,        2438670,         1886040,          702000,          123120,           7776;
  4:       18193357,       44212428,        43896060,        22632480,         6318000,         886464,          46656;
  5:      265474495,      764120994,       928460988,       614544840,       237641040,       53071200,        6205248,        279936;
  6:     3827659537,    12742775760,     18338903856,     14855375808,      7374538080,     2281353984,      424569600,      42550272,       1679616;
  7:    54653958307,   206693614998,    344054945520,    330100269408,    200547573408,    79645011264,    20532185856,    3275251200,     287214336,     10077696;
  8:   774128454229,  3279237498420,   6200808449940,   6881098910400,   4951504041120,  2406570880896,   796450112640,  175990164480,   24564384000,   1914762240,    60466176;
  9: 10890820856071, 51092477979114, 108214837447860, 136417785898680, 113538132021600, 65359853342784, 26472279689856, 7509386776320, 1451918856960, 180138816000, 12637430784, 362797056;

The row polynomials satisfy (6*n+19+6*x)*P(n,x) = (78*n+325+(36*n+228)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 325 + 228*x + 36*x^2.

P[0, x_] := 325 + 228*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((78*n + 325 + (36*n + 228)*x + 36*x^2)*P[n - 1, x])/(6*n + 19 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 325 + 228*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((78*n + 325 + (36*n + 228)*x + 36*x^2)*v[i - 1])/(6*n + 19 + 6*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_A013609 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

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.