Real-rooted atlas

A272867 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 8\,x^{2} + 12\,n x + 16\,n x^{2}\right)\,P_{n}(x) = \left(18 + 144\,x + 384\,x^{2} + 384\,x^{3} + 128\,x^{4} + 72\,n x + 384\,n x^{2} + 576\,n x^{3} + 256\,n x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
164848102
2365762304307212807268
321651843628811059216128011059228672452824
41296414724147201990656525312079626246881280314572858982426280720
5777631104040435202654208010229760024684134438191104037748736023003136078643200115343361459650656
64665622394883583180829859840015178752005064228864114944901121800614707219428802560141557760006643777536181193932821810380878677856832
727993615676416297851904300987187219020718080812665405442447657533445313512079368391968686089631589990407935161794564566087106561740468387843946001203240265318404149742040448

All 15 rows

Technical details
Node id
n-4670037a8155e6a9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1d2b3853bdcde690

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,        48,          48;
  2:       36,       576,        2304,         3072,          1280;
  3:      216,      5184,       36288,       110592,        161280,         110592,          28672;
  4:     1296,     41472,      414720,      1990656,       5253120,        7962624,        6881280,         3145728,          589824;
  5:     7776,    311040,     4043520,     26542080,     102297600,      246841344,      381911040,       377487360,       230031360,         78643200,         11534336;
  6:    46656,   2239488,    35831808,    298598400,    1517875200,     5064228864,    11494490112,     18006147072,     19428802560,      14155776000,       6643777536,       1811939328,        218103808;
  7:   279936,  15676416,   297851904,   3009871872,   19020718080,    81266540544,   244765753344,    531351207936,    839196868608,     963158999040,     793516179456,     456608710656,     174046838784,      39460012032,      4026531840;
  8:  1679616, 107495424,  2364899328,  28092137472,  212363182080,  1107632848896,  4170121740288,  11635368394752,  24416845037568,   38784561315840,   46599455637504,   42008001380352,   27920777084928,   13258564042752,   4252017623040,    824633720832,    73014444032;
  9: 10077696, 725594112, 18139852800, 247669456896, 2180007198720, 13436068036608, 60791385489408, 207822193164288, 546441378398208, 1116530603458560, 1780994122186752, 2216770060419072, 2140886041362432, 1585345443397632, 881907109724160, 356241767399424, 98569499443200, 16698832846848, 1305670057984;
%F Formula
The row polynomials satisfy (3+12*n*x+(16*n-8)*x^2)*P(n,x) = (18+(72*n+144)*x+(384*n+384)*x^2+(576*n+384)*x^3+(256*n+128)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((18 + (72*n + 144)*x + (384*n + 384)*x^2 + (576*n + 384)*x^3 + (256*n + 128)*x^4)*P[n - 1, x])/(3 + 12*n*x + (16*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((18 + (72*n + 144)*x + (384*n + 384)*x^2 + (576*n + 384)*x^3 + (256*n + 128)*x^4)*v[i - 1])/(3 + 12*n*x + (16*n - 8)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.