A395972 → D

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Submission text ?

Triangle begins:
   1:     16,       16;
   2:     72,      288,        144;
   3:    256,     2304,       4608,        1536;
   4:    800,    12800,      57600,       76800,        19200;
   5:   2304,    57600,     460800,     1382400,      1382400,        276480;
   6:   6272,   225792,    2822400,    15052800,     33868800,      27095040,       4515840;
   7:  16384,   802816,   14450688,   120422400,    481689600,     867041280,     578027520,       82575360;
   8:  41472,  2654208,   65028096,   780337152,   4877107200,   15606743040,   23410114560,    13377208320,     1672151040;
   9: 102400,  8294400,  265420800,  4335206400,  39016857600,  195084288000,  520224768000,   668860416000,   334430208000,   37158912000;
  10: 247808, 24780800, 1003622400, 21410611200, 262279987200, 1888415907840, 7868399616000, 17984913408000, 20233027584000, 8992456704000, 899245670400;

The row polynomials satisfy n^2*P(n,x) = (2*n^2+4*n+2+(4*n^3+6*n^2-2)*x)*P(n-1,x) - (2*n^2+4*n+2)*x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 16 + 16*x.

P[1, x_] := 16 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 4*n + 2 + (4*n^3 + 6*n^2 - 2)*x)*P[n - 1, x] - (2*n^2 + 4*n + 2)*x^2*D[P[n - 1, x], x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 16 + 16*x; for(n = 2, N, my(i = n); v[i] = ((2*n^2 + 4*n + 2 + (4*n^3 + 6*n^2 - 2)*x)*v[i - 1] - (2*n^2 + 4*n + 2)*x^2*deriv(v[i - 1], x))/(n^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).?
    • 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

Related ?

Same fingerprint ?

• A271703 → D+1
• A105278 → θ+1
• A144084 → D
• Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
• A395972 → reverse
• A130534 → basis A008277 → θ+1
• A271703 → D
• A132393 → basis A008277
• A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

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.