Real-rooted atlas

A136124 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
2125648116
3603765762561268
4360273657124608128014696
52520220325640062720320006144181816
620160196416583392802560569600202752286722403552
7181440192902464290241033446491392004558848120422413107233907296
81814400207417607586438413763276814306432089456640333168646815744589824509296704
9199584002426745609589587841918570496226187136016707317767839498242272788483715891226214408123774400
10239500800307176192012963552768281372928003673530880030905763840172041461766311116800146866176019660800011534336137245248000
113113510400418489113601869567575044349237544966182454784005781044121603678807982081606920437764749262848090832896001015021568503316482449406937600
124358914560061079284224028684880732167086035269632108300554700801106104006656078481517649923931429404672138801099571233824440320054176776192513382809621810380846065366144000
13653837184000951060579840046692078151680121589132101632197881008399360217899867254784169340463452160948487062814723851159725670411242603806720230092701696031341517209625518145536939524096910810699545600
141046139494400015740039024640080413688521728021944505304350723774041794897920443622671639347237263134624235522291707133427712104300473437388835104431538176086282289020928150550515548161766422740992124956704768402653184018892020104294400
Technical details
Node id
n-6ea588a80c97f398 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-466d95fa7350938a

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:        3,         8;
  2:       12,        56,        48;
  3:       60,       376,       576,        256;
  4:      360,      2736,      5712,       4608,       1280;
  5:     2520,     22032,     56400,      62720,      32000,       6144;
  6:    20160,    196416,    583392,     802560,     569600,     202752,     28672;
  7:   181440,   1929024,   6429024,   10334464,    9139200,    4558848,   1204224,    131072;
  8:  1814400,  20741760,  75864384,  137632768,  143064320,   89456640,  33316864,   6815744,   589824;
  9: 19958400, 242674560, 958958784, 1918570496, 2261871360, 1670731776, 783949824, 227278848, 37158912, 2621440;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(8*n+8)*x+16*x^2)*P(n-2,x) + (2*n+3+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + 1 + (8*n + 8)*x + 16*x^2)*P[n - 2, x] + (2*n + 3 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n + 1 + (8*n + 8)*x + 16*x^2)*v[i - 2] + (2*n + 3 + 8*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)

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.