Real-rooted atlas

A038305 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 361 + 456\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0361456144961
16859129968208172829791
213032132923231190413132820736923521
32476099781926098769606238080196992024883228629151
447045881178279128281493360237047040112285440283668482985984887503681
5893871739395185400474877233767881814080497798784018863953923971358723583180827512614111
6169835630418581168694418968899219223960714803218916353792095577366528301823262725446434816429981696852891037441
732268769777918342248084284633831094976682880371891264693929968644085932419072172039259750446567017676873526870016515978035226439622160671

All 15 rows

Technical details
Node id
n-e2ddb0b94b67e233 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-b68b92f4c09b2b20

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,             456,              144;
  1:            6859,           12996,             8208,             1728;
  2:          130321,          329232,           311904,           131328,            20736;
  3:         2476099,         7819260,          9876960,          6238080,          1969920,           248832;
  4:        47045881,       178279128,        281493360,        237047040,        112285440,         28366848,          2985984;
  5:       893871739,      3951854004,       7487723376,       7881814080,       4977987840,       1886395392,        397135872,         35831808;
  6:     16983563041,     85811686944,     189688992192,     239607148032,     189163537920,      95577366528,      30182326272,       5446434816,       429981696;
  7:    322687697779,   1834224808428,    4633831094976,    6828803718912,    6469392996864,    4085932419072,    1720392597504,     465670176768,     73526870016,      5159780352;
  8:   6131066257801,  38722523733480,  110053488505680,  185353243799040,  204864111567360,  155265431924736,   81718648381440,   29492444528640,   6985052651520,    980358266880,    61917364224;
  9: 116490258898219, 809300746029732, 2555686566409680, 4842353494249920, 6116657045368320, 5408412545378304, 3415839502344192, 1540980226621440, 486625334722560, 102447438888960, 12940729122816, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (19+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 456*x + 144*x^2.
%t Mathematica
P[0, x_] := 361 + 456*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 12*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] = 361 + 456*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 12*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.