Real-rooted atlas

A034870 → basis A013609 → basis A013609

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 32\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116321664
22561024153610242564096
3409624576614408192061440245764096262144
465536524288183500836700164587520367001618350085242886553616777216
510485761048576047185920125829120220200960264241152220200960125829120471859201048576010485761073741824
6167772162013265921107296256369098752083047219201328755507215502147584132875550728304721920369098752011072962562013265921677721668719476736
726843545637580963842442762649697710505984268703891456537407782912806111674368921270484992806111674368537407782912268703891456977105059842442762649637580963842684354564398046511104

All 15 rows

Technical details
Node id
n-c352f14299f42ecc · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-2cd9f46ca2d53115

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: yes
  • γ-positive (gamma_positive): yes
  • 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:          16,            32,             16;
  2:         256,          1024,           1536,           1024,             256;
  3:        4096,         24576,          61440,          81920,           61440,           24576,             4096;
  4:       65536,        524288,        1835008,        3670016,         4587520,         3670016,          1835008,           524288,            65536;
  5:     1048576,      10485760,       47185920,      125829120,       220200960,       264241152,        220200960,        125829120,         47185920,         10485760,          1048576;
  6:    16777216,     201326592,     1107296256,     3690987520,      8304721920,     13287555072,      15502147584,      13287555072,       8304721920,       3690987520,       1107296256,        201326592,         16777216;
  7:   268435456,    3758096384,    24427626496,    97710505984,    268703891456,    537407782912,     806111674368,     921270484992,     806111674368,     537407782912,     268703891456,      97710505984,      24427626496,      3758096384,       268435456;
  8:  4294967296,   68719476736,   515396075520,  2405181685760,   7816840478720,  18760417148928,   34394098106368,   49134425866240,   55276229099520,   49134425866240,   34394098106368,   18760417148928,    7816840478720,   2405181685760,    515396075520,    68719476736,     4294967296;
  9: 68719476736, 1236950581248, 10514079940608, 56075093016576, 210281598812160, 588788476674048, 1275708366127104, 2186928627646464, 3007026863013888, 3341140958904320, 3007026863013888, 2186928627646464, 1275708366127104, 588788476674048, 210281598812160, 56075093016576, 10514079940608, 1236950581248, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (16+32*x+16*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 + 32*x + 16*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 + 32*x + 16*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.