A143491 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:          21,          18;
  1:         141,         216,           81;
  2:        1026,        2142,         1458,          324;
  3:        8262,       21150,        19845,         8100,         1215;
  4:       73656,      218772,       253935,       144180,        40095,        4374;
  5:      723384,     2410884,      3269889,      2313360,       901530,      183708,       15309;
  6:     7778160,    28449144,     43547868,     36213156,     17690400,     5082588,      796068,      52488;
  7:    91002960,   359609544,    607047696,    572536188,    330393735,   119593908,    26545806,    3306744,    177147;
  8:  1151910720,  4861332288,   8902815300,   9298625040,   6111735525,  2624544342,   737128350,  130695120,  13286025,   590490;
  9: 15693341760, 70108784064, 137612594196, 156493386720, 114322405725, 56121324336, 18768573747, 4226333760, 613814355, 51963120, 1948617;

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(6*n+18)*x+9*x^2)*P(n-2,x) + (2*n+7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 21 + 18*x, P(1,x) = 141 + 216*x + 81*x^2.

P[0, x_] := 21 + 18*x;
P[1, x_] := 141 + 216*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (6*n + 18)*x + 9*x^2)*P[n - 2, x] + (2*n + 7 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 21 + 18*x; v[2] = 141 + 216*x + 81*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (6*n + 18)*x + 9*x^2)*v[i - 2] + (2*n + 7 + 6*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 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_A013610 of Unsigned 2-Stirling numbers of the first kind.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A136124 → basis A013610 → D

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.