Real-rooted atlas

A038321 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 7\,x + 7\,n\right)\,P_{n}(x) = \left(360 + 546\,x + 147\,x^{2} + 126\,n + 147\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1080 + 1638\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0108016384413159
12624468040515979261155142
25948642204496285768014447161944817296237
31291204862460720115736040100018800379237954084101333135504
427209779216269180483935025360486091368031505922009556796348576612114886992835
556120169603999837542411957847652819281624264017863857774092627410680234141510331801088541654486339546
6113872925952942818849280335986353561666963946855688098282190880600225621206425935674990405619396247923782285936128406818382553
72281131839232215219830049288909638125696021167140274380831640464889308830611506681526418907107068001670026322474080132758236357117942800465811220258111389668

All 15 rows

Technical details
Node id
n-9848ee772d97c947 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d56b748473677c64

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:            1080,              1638,               441;
  1:           26244,             68040,             51597,               9261;
  2:          594864,           2204496,           2857680,            1444716,             194481;
  3:        12912048,          62460720,         115736040,          100018800,           37923795,            4084101;
  4:       272097792,        1626918048,        3935025360,         4860913680,         3150592200,          955679634,           85766121;
  5:      5612016960,       39998375424,      119578476528,       192816242640,       178638577740,        92627410680,        23414151033,         1801088541;
  6:    113872925952,      942818849280,     3359863535616,      6696394685568,      8098282190880,      6002256212064,      2593567499040,       561939624792,        37822859361;
  7:   2281131839232,    21521983004928,    89096381256960,    211671402743808,    316404648893088,    306115066815264,    189071070680016,     70026322474080,     13275823635711,      794280046581;
  8:  45225918203904,   479037686238720,  2259808215517440,   6236746687987200,  11112748644049920,  13288995253509696,  10714027338534240,   5672132120400480,   1838190964944600,   309769218166590,   16679880978201;
  9: 889046339429376, 10447187105101824, 55328852760572160, 174005232594842880, 360172121231260800, 513408987355106304, 511626317260123296, 353562902171629920, 163782814976563860, 47180234766911400, 7155668939648229, 350277500542221;
%F Formula
The row polynomials satisfy (7*n+13+7*x)*P(n,x) = (126*n+360+(147*n+546)*x+147*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1080 + 1638*x + 441*x^2.
%t Mathematica
P[0, x_] := 1080 + 1638*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((126*n + 360 + (147*n + 546)*x + 147*x^2)*P[n - 1, x])/(7*n + 13 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1080 + 1638*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((126*n + 360 + (147*n + 546)*x + 147*x^2)*v[i - 1])/(7*n + 13 + 7*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.