Real-rooted atlas

A272866 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(16 + 20\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11620541
2256640560200251681
340961536023040176007200150012568921
46553632768069632081920058240025600068000100006252825761
5104857665536001802240028672000291840001984000091200002800000550000625003125115856201
61677721612582912042467328085196800011304960001044480000688640000326400000110400000260000004050000375000156254750104241
7268435456234881024093952409602275409920037250662400435814400003756032000024217600000117376000004256000000113680000021700000028000000218750078125194754273881

All 15 rows

Technical details
Node id
n-bc4e6b211c5c1593 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d98c1510b99a0b49

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:          16,           20,             5;
  2:         256,          640,           560,            200,             25;
  3:        4096,        15360,         23040,          17600,           7200,           1500,            125;
  4:       65536,       327680,        696320,         819200,         582400,         256000,          68000,          10000,            625;
  5:     1048576,      6553600,      18022400,       28672000,       29184000,       19840000,        9120000,        2800000,         550000,          62500,           3125;
  6:    16777216,    125829120,     424673280,      851968000,     1130496000,     1044480000,      688640000,      326400000,      110400000,       26000000,        4050000,        375000,         15625;
  7:   268435456,   2348810240,    9395240960,    22754099200,    37250662400,    43581440000,    37560320000,    24217600000,    11737600000,     4256000000,     1136800000,     217000000,      28000000,      2187500,       78125;
  8:  4294967296,  42949672960,  198642237440,   563714457600,  1098068787200,  1556086784000,  1658847232000,  1356595200000,   859955200000,   423936000000,   161996800000,   47488000000,   10472000000,   1680000000,   185000000,    12500000,     390625;
  9: 68719476736, 773094113280, 4058744094720, 13207024435200, 29836600934400, 49677336576000, 63153635328000, 62662901760000, 49185423360000, 30765056000000, 15370444800000, 6119424000000, 1927296000000, 473760000000, 88920000000, 12300000000, 1181250000, 70312500, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (16+20*x+5*x^2)*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[(16 + 20*x + 5*x^2)*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] = (16 + 20*x + 5*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.