Real-rooted atlas

A038325 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(22 + 66\,x + 66\,x^{2} + 22\,x^{3} + 66\,n x + 110\,n x^{2} + 44\,n x^{3} + 22\,n^{2} x^{2} + 22\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12288110
2484387243568712
310648127776287496170368596288
423425637480961264982414992384585640037480960
551536321030726404638268808245811206442040001855307522226369024
61133799042721117696153062870403628156928042517464000244900592645555615296126985492480
72494357888698420208644714336408321396840417280218256315200018857345633288555647555841596389048327024111812608

All 15 rows

Technical details
Node id
n-800be4c35d3e0bbc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:            22,             88;
  2:           484,           3872,            4356;
  3:         10648,         127776,          287496,           170368;
  4:        234256,        3748096,        12649824,         14992384,          5856400;
  5:       5153632,      103072640,       463826880,        824581120,        644204000,        185530752;
  6:     113379904,     2721117696,     15306287040,      36281569280,      42517464000,      24490059264,       5555615296;
  7:    2494357888,    69842020864,    471433640832,    1396840417280,    2182563152000,    1885734563328,     855564755584,     159638904832;
  8:   54875873536,  1756027953152,  13828720131072,   49168782688256,   96032778688000,  110629761048576,   75289698491392,   28096447250432,   4444945756416;
  9: 1207269217792, 43461691840512, 391155226564608, 1622569828712448, 3802898036044800, 5476173171904512, 4969120100431872, 2781548277792768, 880099259770368, 120726921779200;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (22+(66*n+66)*x+(22*n^2+110*n+66)*x^2+(22*n^2+44*n+22)*x^3)*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[((22 + (66*n + 66)*x + (22*n^2 + 110*n + 66)*x^2 + (22*n^2 + 44*n + 22)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*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] = ((22 + (66*n + 66)*x + (22*n^2 + 110*n + 66)*x^2 + (22*n^2 + 44*n + 22)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*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.