Real-rooted atlas

A094638 → basis A013609 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
228120128276
32801872416030729384
436403329611398417305698304422280
558240678336315558473282568495104393216023647680
6110656015683904925162242907048965131632644824760321887436801584394560
72434432040701324829136555521157641625627569065984393516154883117101875210569646080123582775680

All 15 rows

Technical details
Node id
n-3b859fdd8b85bbb3 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-d6f36cebd589d97b

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           4,            8;
  2:          28,          120,           128;
  3:         280,         1872,          4160,           3072;
  4:        3640,        33296,        113984,         173056,          98304;
  5:       58240,       678336,       3155584,        7328256,        8495104,         3932160;
  6:     1106560,     15683904,      92516224,      290704896,      513163264,       482476032,       188743680;
  7:    24344320,    407013248,    2913655552,    11576416256,    27569065984,     39351615488,     31171018752,     10569646080;
  8:   608608000,  11733367680,   98890236672,   475884361728,  1430117289984,   2748210610176,   3297778860032,   2259186352128,    676457349120;
  9: 17041024000, 372354071040, 3613729099776, 20444859168768, 74306958163968, 179918341963776, 290208972013568, 300697295781888, 181602223128576, 48704929136640;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+8*n*x)*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[(3*n + 1 + 8*n*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; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 1 + 8*n*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

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.