Real-rooted atlas

A139548 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + x^{2} + 6\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 12\,x^{2} + 8\,x^{3} + 2\,x^{4} + 12\,n x + 32\,n x^{2} + 28\,n x^{3} + 8\,n x^{4} + 8\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12161836
2464216256100640
3819210802560300017283928960
41651240321433628000322562195281921296110592
532128012960614401680002903043292802457601166403200038721261568
664307238016225280792000182476828976643244032256608014080005111041105921081613631488
712871681048327454723203200922521618834816281149443113510425625600155034886709248196851235123228800141557760

All 15 rows

Technical details
Node id
n-011c3414b63bdcbb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-561a7d876fc231a6

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:   1;
  1:   2,    16,     18;
  2:   4,    64,    216,     256,      100;
  3:   8,   192,   1080,    2560,     3000,      1728,       392;
  4:  16,   512,   4032,   14336,    28000,     32256,     21952,       8192,       1296;
  5:  32,  1280,  12960,   61440,   168000,    290304,    329280,     245760,     116640,      32000,       3872;
  6:  64,  3072,  38016,  225280,   792000,   1824768,   2897664,    3244032,    2566080,    1408000,     511104,     110592,      10816;
  7: 128,  7168, 104832,  745472,  3203200,   9225216,  18834816,   28114944,   31135104,   25625600,   15503488,    6709248,    1968512,    351232,     28800;
  8: 256, 16384, 276480, 2293760, 11648000,  40255488, 100452352,  187432960,  266872320,  292864000,  248055808,  161021952,   78740480,  28098560,   6912000,   1048576,    73984;
  9: 512, 36864, 705024, 6684672, 39168000, 157925376, 465733632, 1042808832, 1814731776, 2489344000, 2710895616, 2346319872, 1606305792, 859815936, 352512000, 106954752, 22639104, 2985984, 184832;
%F Formula
The row polynomials satisfy (1+(6*n-4)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (2+(12*n+8)*x+(8*n^2+32*n+12)*x^2+(16*n^2+28*n+8)*x^3+(8*n^2+8*n+2)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (12*n + 8)*x + (8*n^2 + 32*n + 12)*x^2 + (16*n^2 + 28*n + 8)*x^3 + (8*n^2 + 8*n + 2)*x^4)*P[n - 1, x])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (12*n + 8)*x + (8*n^2 + 32*n + 12)*x^2 + (16*n^2 + 28*n + 8)*x^3 + (8*n^2 + 8*n + 2)*x^4)*v[i - 1])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.