Real-rooted atlas

A038319 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(16 + 52\,x + 36\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 1152\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0256115212962704
14096276486220846656140608
2655365898241990656298598416796167311616
31048576117964805308416011943936013436928060466176380204032
4167772162264924161274019840382205952064497254405804752896217678233619770609664
5268435456422785843228538044416107017666560240789749760325066162176243799621632783641640961028071702528
642949672967730941132860881161420827396522639367705271992320138694895861761560317578444810030613004288282110990745653459728531456
768719476736139156940390412524124635136657516543344642219118333788164993016251023367489524376535047222041363087364062398266736641015599566684162779905883635712

All 15 rows

Technical details
Node id
n-a3266ffb1baebbd1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-12e1fe8c90bb4450

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:            256,            1152,             1296;
  1:           4096,           27648,            62208,             46656;
  2:          65536,          589824,          1990656,           2985984,            1679616;
  3:        1048576,        11796480,         53084160,         119439360,          134369280,           60466176;
  4:       16777216,       226492416,       1274019840,        3822059520,         6449725440,         5804752896,          2176782336;
  5:      268435456,      4227858432,      28538044416,      107017666560,       240789749760,       325066162176,        243799621632,         78364164096;
  6:     4294967296,     77309411328,     608811614208,     2739652263936,      7705271992320,     13869489586176,      15603175784448,      10030613004288,       2821109907456;
  7:    68719476736,   1391569403904,   12524124635136,    65751654334464,    221911833378816,    499301625102336,     748952437653504,     722204136308736,     406239826673664,     101559956668416;
  8:  1099511627776,  24739011624960,  250482492702720,  1502894956216320,   5917648890101760,  15977652003274752,   29958097506140160,   38517553936465920,   32499186133893120,   16249593066946560,   3656158440062976;
  9: 17592186044416, 435406604599296, 4898324301742080, 33063689036759040, 148786600665415680, 468677792096059392, 1054525032216133632, 1694772373204500480, 1906618919855063040, 1429964189891297280, 643483885451083776, 131621703842267136;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (16+52*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 1152*x + 1296*x^2.
%t Mathematica
P[0, x_] := 256 + 1152*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + 52*x + 36*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] = 256 + 1152*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16 + 52*x + 36*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

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.