Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(16 + 24\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11624949
2256768864432812401
34096184323456034560194405832729117649
4655363932161032192154828814515208709123265926998465615764801
510485767864320265420805308416069672960627056643919104016796160472392078732059049282475249
6167772161509949446228541441557135360262766592031531991042759049216177367449683140992027713664062355744850305653144113841287201
726843545628185722881374053990441221619712850195906561275293859841434705592321229747650568070218956840351094784151316605444126816512773778096892820884782969678223072849

All 15 rows

Technical details
Node id
n-31f600b3d147eeab · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7cfe8b61bc182db1

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          16,           24,             9;
  2:         256,          768,           864,            432,             81;
  3:        4096,        18432,         34560,          34560,          19440,            5832,             729;
  4:       65536,       393216,       1032192,        1548288,        1451520,          870912,          326592,           69984,            6561;
  5:     1048576,      7864320,      26542080,       53084160,       69672960,        62705664,        39191040,        16796160,         4723920,          787320,           59049;
  6:    16777216,    150994944,     622854144,     1557135360,     2627665920,      3153199104,      2759049216,      1773674496,       831409920,       277136640,        62355744,        8503056,         531441;
  7:   268435456,   2818572288,   13740539904,    41221619712,    85019590656,    127529385984,    143470559232,    122974765056,     80702189568,     40351094784,     15131660544,     4126816512,      773778096,       89282088,       4782969;
  8:  4294967296,  51539607552,  289910292480,  1014686023680,  2473297182720,   4451934928896,   6121410527232,   6558654136320,   5533864427520,   3689242951680,   1936852549632,   792348770304,   247608990720,    57140536320,    9183300480,    918330048,     43046721;
  9: 68719476736, 927712935936, 5914169966592, 23656679866368, 66534412124160, 139722265460736, 227048681373696, 291919733194752, 301042224857088, 250868520714240, 169336251482112, 92365228081152, 40409787285504, 13988003291136, 3746786595840, 749357319168, 105378373008, 9298091736, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (16+24*x+9*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 + 24*x + 9*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 + 24*x + 9*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.