Real-rooted atlas

A137320 → basis A013609 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 16\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
161622
242208256506
333623365376409612144
43024264008576012288065536303600
53024031238412800002600960262144010485767893600
633264039200641907814449090560704512005347737616777216213127200
73991680523630082916587528943370241630863360176894771210569646082684354565967561600

All 15 rows

Technical details
Node id
n-25622692aaaa49c0 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-a8a66b2e4166f02a

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:         6,          16;
  2:        42,         208,         256;
  3:       336,        2336,        5376,         4096;
  4:      3024,       26400,       85760,       122880,        65536;
  5:     30240,      312384,     1280000,      2600960,      2621440,       1048576;
  6:    332640,     3920064,    19078144,     49090560,     70451200,      53477376,      16777216;
  7:   3991680,    52363008,   291658752,    894337024,   1630863360,    1768947712,    1056964608,    268435456;
  8:  51891840,   744585984,  4629371904,  16292921344,  35510616064,   49090134016,   42043703296,  20401094656,   4294967296;
  9: 726485760, 11254473216, 76724582400, 302170849280, 757835366400, 1255431733248, 1374053990400, 958314577920, 386547056640, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (n+5+16*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[(n + 5 + 16*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] = (n + 5 + 16*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.