Real-rooted atlas

A038319 → basis A013611 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-576 + 80\,x\right)\,P_{n-2}(x) + \left(52 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1296 + 1840\,x + 400\,x^{2}\)
\(P_{1}(x) = 46656 + 112960\,x + 59200\,x^{2} + 8000\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0129618404003536
146656112960592008000226816
2167961658508805254400163200016000014576896
360466176276505600365184000190080000416000003200000937035776
42176782336123518771202196121600016668160000600320000010112000006400000060236435456
578364164096531403325440120079523840012256972800006367744000001741312000002380800000012800000003872253607936
628211099074562225971757056061407908659200799082004480005550182400000021688115200000476467200000054784000000025600000000248924987785216
710155995666841691410832949248029892598104064004773446582272000421553258496000021884608512000006817423360000001249484800000001239040000000051200000000016001961330999296

All 15 rows

Technical details
Node id
n-387a2884c2e42811 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-427fb160a6bb2f85

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1296,                1840,                 400;
  1:              46656,              112960,               59200,                 8000;
  2:            1679616,             5850880,             5254400,              1632000,               160000;
  3:           60466176,           276505600,           365184000,            190080000,             41600000,              3200000;
  4:         2176782336,         12351877120,         21961216000,          16668160000,           6003200000,           1011200000,            64000000;
  5:        78364164096,        531403325440,       1200795238400,        1225697280000,         636774400000,         174131200000,         23808000000,          1280000000;
  6:      2821109907456,      22259717570560,      61407908659200,       79908200448000,       55501824000000,       21688115200000,       4764672000000,        547840000000,        25600000000;
  7:    101559956668416,     914108329492480,    2989259810406400,     4773446582272000,     4215532584960000,     2188460851200000,     681742336000000,     124948480000000,     12390400000000,      512000000000;
  8:   3656158440062976,   36968923738931200,  140133498748928000,   266889927720960000,   289100231475200000,   190058407526400000,   78210416640000000,   20197785600000000,   3172352000000000,   276480000000000,   10240000000000;
  9: 131621703842267136, 1477105601971486720, 6377635425288192000, 14170581769912320000, 18324739548774400000, 14741730977382400000, 7650503098368000000, 2597062246400000000, 571777024000000000, 78520320000000000, 6103040000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = -(576-80*x)*P(n-2,x) + (52+20*x)*P(n-1,x) for n >= 2, with P(0,x) = 1296 + 1840*x + 400*x^2, P(1,x) = 46656 + 112960*x + 59200*x^2 + 8000*x^3.
%t Mathematica
P[0, x_] := 1296 + 1840*x + 400*x^2;
P[1, x_] := 46656 + 112960*x + 59200*x^2 + 8000*x^3;
P[n_, x_] := P[n, x] = Expand[-(576 - 80*x)*P[n - 2, x] + (52 + 20*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] = 1296 + 1840*x + 400*x^2; v[2] = 46656 + 112960*x + 59200*x^2 + 8000*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(576 - 80*x)*v[i - 2] + (52 + 20*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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

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.