Real-rooted atlas

A038226 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 361 + 608\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03616082561225
168591732814592409642875
2130321438976554496311296655361500625
324760991042568017559040147865606225920104857652521875
447045881237705504500432640561889280354877440119537664167772161838265625
589387173952691386721331150822418682818560157328998407949254656223136972826843545664339296875
6169835630411144155825923372248750085679576842245978501939204027622359041695840993284080218931242949672962251875390625
732268769777924456330779048237921946624161867940003842044647663206417218085584896966629366169634885871861767344394076166871947673678815638671875

All 15 rows

Technical details
Node id
n-e75956a967cba1f2 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4b6c87ac223a4827

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,              608,              256;
  1:            6859,            17328,            14592,              4096;
  2:          130321,           438976,           554496,            311296,             65536;
  3:         2476099,         10425680,         17559040,          14786560,           6225920,           1048576;
  4:        47045881,        237705504,        500432640,         561889280,         354877440,         119537664,          16777216;
  5:       893871739,       5269138672,      13311508224,       18682818560,       15732899840,        7949254656,        2231369728,         268435456;
  6:     16983563041,     114415582592,     337224875008,      567957684224,      597850193920,      402762235904,      169584099328,       40802189312,       4294967296;
  7:    322687697779,    2445633077904,    8237921946624,    16186794000384,    20446476632064,    17218085584896,     9666293661696,     3488587186176,     734439407616,      68719476736;
  8:   6131066257801,   51630031644640,  195650646232320,   439355837153280,   647471760015360,   654287252226048,   459148948930560,   220943855124480,   69771743723520,   13056700579840,   1099511627776;
  9: 116490258898219, 1079067661372976, 4543442784728320, 11478171245629440, 19331656834744320, 22791005952540672, 19192426065297408, 11544316430254080, 4860764812738560, 1364425210593280, 229797930205184, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (19+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 608*x + 256*x^2.
%t Mathematica
P[0, x_] := 361 + 608*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 16*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 + 608*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 16*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.