Real-rooted atlas

A038281 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 165\,x + 36\,x^{2} + 99\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 3267\,x + 1683\,x^{2} + 243\,x^{3} + 1089\,n x + 891\,n x^{2} + 162\,n x^{3} + 99\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 264\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
012126481466
11331435626734328792
2146416388858806190082025158368
3161051878460107811052272011137587482760464
4177156111595672177888151149984036753755773683572146944352
51948717114881112427394775122137192094334625222286682750517139968783071744
621435888118707684164017900348389614579220753617506520409281210227481231718453144112860447488
7235794769123150759148568245906366428640556841092162650161380129683993750684609700608526126591968300208507910912

All 15 rows

Technical details
Node id
n-03bdbf755fc5d2ed · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-83385ff7049ec979

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:          121,           264,             81;
  1:         1331,          4356,           2673,            432;
  2:        14641,         63888,          58806,          19008,           2025;
  3:       161051,        878460,        1078110,         522720,         111375,          8748;
  4:      1771561,      11595672,       17788815,       11499840,        3675375,        577368,         35721;
  5:     19487171,     148811124,      273947751,      221371920,       94334625,      22228668,       2750517,       139968;
  6:    214358881,    1870768416,     4017900348,     3896145792,     2075361750,     652040928,     121022748,     12317184,       531441;
  7:   2357947691,   23150759148,    56824590636,    64286405568,    41092162650,   16138012968,    3993750684,    609700608,     52612659,     1968300;
  8:  25937424601,  282953722920,   781338121245,  1010214944640,   753356315250,  355036285296,  109828143810,  22355688960,   2893696245,   216513000,   7144929;
  9: 285311670611, 3423740047332, 10504656963405, 15279501037680, 13022302020750, 7159898420136, 2657841080202, 676259591040, 116712415215, 13099036500, 864536409, 25509168;
%F Formula
The row polynomials satisfy (121+(99*n+165)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (1331+(1089*n+3267)*x+(99*n^2+891*n+1683)*x^2+(27*n^2+162*n+243)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 264*x + 81*x^2.
%t Mathematica
P[0, x_] := 121 + 264*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (1089*n + 3267)*x + (99*n^2 + 891*n + 1683)*x^2 + (27*n^2 + 162*n + 243)*x^3)*P[n - 1, x])/(121 + (99*n + 165)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 264*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (1089*n + 3267)*x + (99*n^2 + 891*n + 1683)*x^2 + (27*n^2 + 162*n + 243)*x^3)*v[i - 1])/(121 + (99*n + 165)*x + (9*n^2 + 36*n + 36)*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.