Real-rooted atlas

A038226 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(9 + 120\,x + 256\,x^{2} + 72\,n x + 256\,n x^{2} + 64\,n^{2} x^{2}\right)\,P_{n}(x) = \left(27 + 648\,x + 3264\,x^{2} + 4608\,x^{3} + 216\,n x + 1728\,n x^{2} + 3072\,n x^{3} + 192\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 9 + 192\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
09192576777
1278645184819214267
28134563110498304102400235345
324312960155520737280153600011796483621651
472946656699840442368013824000212336641284505653073625
5218716329629393282322432096768000222953472269746176134217728750014507
6656155987211757312111476736580608000178362777632369541123221225472135895449610305170337
71968318895684534963250164531231352832001203948748829132587008434865438723669177139213421772800138456349955

All 15 rows

Technical details
Node id
n-e226c8909291af75 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-476cf4c2bd2dc965

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      9,      192,       576;
  1:     27,      864,      5184,       8192;
  2:     81,     3456,     31104,      98304,      102400;
  3:    243,    12960,    155520,     737280,     1536000,      1179648;
  4:    729,    46656,    699840,    4423680,    13824000,     21233664,      12845056;
  5:   2187,   163296,   2939328,   23224320,    96768000,    222953472,     269746176,     134217728;
  6:   6561,   559872,  11757312,  111476736,   580608000,   1783627776,    3236954112,    3221225472,    1358954496;
  7:  19683,  1889568,  45349632,  501645312,  3135283200,  12039487488,   29132587008,   43486543872,   36691771392,   13421772800;
  8:  59049,  6298560, 170061120, 2149908480, 15676416000,  72236924928,  218494402560,  434865438720,  550376570880,  402653184000,  129922760704;
  9: 177147, 20785248, 623557440, 8868372480, 73903104000, 397303087104, 1442063056896, 3587639869440, 6054142279680, 6643777536000, 4287451103232, 1236950581248;
%F Formula
The row polynomials satisfy (9+(72*n+120)*x+(64*n^2+256*n+256)*x^2)*P(n,x) = (27+(216*n+648)*x+(192*n^2+1728*n+3264)*x^2+(512*n^2+3072*n+4608)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 192*x + 576*x^2.
%t Mathematica
P[0, x_] := 9 + 192*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((27 + (216*n + 648)*x + (192*n^2 + 1728*n + 3264)*x^2 + (512*n^2 + 3072*n + 4608)*x^3)*P[n - 1, x])/(9 + (72*n + 120)*x + (64*n^2 + 256*n + 256)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 9 + 192*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((27 + (216*n + 648)*x + (192*n^2 + 1728*n + 3264)*x^2 + (512*n^2 + 3072*n + 4608)*x^3)*v[i - 1])/(9 + (72*n + 120)*x + (64*n^2 + 256*n + 256)*x^2)); 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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.