Real-rooted atlas

A038333 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(19 + 52\,x + 33\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 361 + 1254\,x + 1089\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0361125410892704
16859357396207335937140608
21303219053882358774273121211859217311616
32476099215029657469451012973257011266249539135393380204032
4470458814902676022128793535492983766064217622154461434802129146796919770609664
58938717391086759851156625908031163917102195284698124865296685414333171765239877426184429771028071702528
616983563041235982139096143452300345249830799067281081852874487015032060992872130541582306526478003332504140640861824153459728531456
7322687697779504411822317735043347655756142017777341748369993683074554642620607445278744087019147164553869284929092240495873719211464114844019532779905883635712

All 15 rows

Technical details
Node id
n-421ecf8294845ce1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2badd0290dfd7765

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:             361,             1254,              1089;
  1:            6859,            35739,             62073,              35937;
  2:          130321,           905388,           2358774,            2731212,            1185921;
  3:         2476099,         21502965,          74694510,          129732570,          112662495,           39135393;
  4:        47045881,        490267602,        2128793535,         4929837660,         6421762215,         4461434802,          1291467969;
  5:       893871739,      10867598511,       56625908031,       163917102195,       284698124865,       296685414333,        171765239877,         42618442977;
  6:     16983563041,     235982139096,     1434523003452,      4983079906728,     10818528744870,     15032060992872,      13054158230652,       6478003332504,       1406408618241;
  7:    322687697779,    5044118223177,    35043347655756,    142017777341748,    369993683074554,    642620607445278,     744087019147164,     553869284929092,     240495873719211,     46411484401953;
  8:   6131066257801,  106486940267070,   832279506824205,   3854768242133160,  11716466630694210,  24419583082920564,   35344133409490290,   35078388045509160,   22847108003325045,   8818182036371070,   1531578985264449;
  9: 116490258898219, 2225577051581763, 19327379658473205, 100705820325728805, 349820217973584270, 850615477388399646, 1477384776516694122, 1832845775377853610, 1591681857564978135, 921500022800776815, 320100007920269841, 50542106513726817;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (19+52*x+33*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 1254*x + 1089*x^2.
%t Mathematica
P[0, x_] := 361 + 1254*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((19 + 52*x + 33*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] = 361 + 1254*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((19 + 52*x + 33*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.