Real-rooted atlas

A038331 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 49\,x + 32\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 1088\,x + 1024\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0289108810242401
14913277445222432768117649
2835216288641775616222822410485765764801
314198571336336050309120946995208912896033554432282475249
4241375692726125441282882560321978368045455769603422552064107374182413841287201
54103386735406815456305326049289578856448018030788608020364184780812777527705634359738368678223072849
6697575744110504670028869207237836826054489538566130468126720923176376729686887188398084672924418048109951162777633232930569601
711858787649720090181430081512672484147266438948323328187592324677632353114964099072443124660830208357478717980672168225279049728351843720888321628413597910449

All 15 rows

Technical details
Node id
n-adf274fa298bd545 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e79a1f872b1c2f24

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            289,            1088,             1024;
  1:           4913,           27744,            52224,             32768;
  2:          83521,          628864,          1775616,           2228224,            1048576;
  3:        1419857,        13363360,         50309120,          94699520,           89128960,           33554432;
  4:       24137569,       272612544,       1282882560,        3219783680,         4545576960,         3422552064,         1073741824;
  5:      410338673,      5406815456,      30532604928,       95788564480,       180307886080,       203641847808,       127775277056,        34359738368;
  6:     6975757441,    105046700288,     692072378368,     2605448953856,      6130468126720,      9231763767296,      8688718839808,      4672924418048,      1099511627776;
  7:   118587876497,   2009018143008,   15126724841472,    66438948323328,    187592324677632,    353114964099072,    443124660830208,    357478717980672,    168225279049728,     35184372088832;
  8:  2015993900449,  37948120479040,  321442902881280,  1613517316423680,   5315115865866240,  12005908779368448,  18832798085283840,  20257127352238080,  14299148719226880,   5981343255101440,   1125899906842624;
  9: 34271896307633, 709629852958048, 6678869204311040, 37715967271403520, 141989523845283840, 374184156956983296, 704346648389615616, 947020703717130240, 891313603498475520, 559255594351984640, 210543282579570688, 36028797018963968;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (17+49*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1088*x + 1024*x^2.
%t Mathematica
P[0, x_] := 289 + 1088*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 49*x + 32*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1088*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 49*x + 32*x^2)*v[i - 1])/(1 + x)); 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.