Real-rooted atlas

A013612 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(21 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 441 + 630\,x + 225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04416302251296
192611984514175337546656
2194481555660595350283500506251679616
34084101145860752083725014883750531562575937560466176
48576612136756909065637337562511750033488437595681250113906252176782336
5180108854190054427051929737722522973068125164093343757032571875167442187517085937578364164096
6378228593612161306249205403265623007718950890006891920437503938240250001406514375002870437500025628906252821109907456
77942800465815106086013735145888171821002431469530350026051459253750186081851812508861040562500271256343750048438632812538443359375101559956668416

All 15 rows

Technical details
Node id
n-1d9325b8d0810008 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f96e0317bce1f626

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:             441,              630,              225;
  1:            9261,            19845,            14175,              3375;
  2:          194481,           555660,           595350,            283500,             50625;
  3:         4084101,         14586075,         20837250,          14883750,           5315625,            759375;
  4:        85766121,        367569090,        656373375,         625117500,         334884375,          95681250,          11390625;
  5:      1801088541,       9005442705,      19297377225,       22973068125,       16409334375,        7032571875,        1674421875,         170859375;
  6:     37822859361,     216130624920,     540326562300,      771895089000,      689192043750,      393824025000,      140651437500,       28704375000,       2562890625;
  7:    794280046581,    5106086013735,   14588817182100,    24314695303500,    26051459253750,    18608185181250,     8861040562500,     2712563437500,     484386328125,     38443359375;
  8:  16679880978201,  119142006987150,  382956451030125,   729440859105000,   911801073881250,   781543777612500,   465204629531250,   189879440625000,   50860564453125,   8073105468750,    576650390625;
  9: 350277500542221, 2752180361403165, 9829215576439875, 21062604806656875, 30089435438081250, 30089435438081250, 21492453884343750, 10965537696093750, 3916263462890625, 932443681640625, 133206240234375, 8649755859375;
%F Formula
The row polynomials satisfy P(n,x) = (21+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 630*x + 225*x^2.
%t Mathematica
P[0, x_] := 441 + 630*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 15*x)*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] = 441 + 630*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 15*x)*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. [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.