Real-rooted atlas

A038301 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(20 + 84\,x + 64\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 2560\,x + 4096\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0400256040967056
1800076800245760262144592704
216000020480009830400209715201677721649787136
33200000512000003276800001048576000167772160010737418244182119424
464000000122880000098304000004194304000010066329600012884901888068719476736351298031616
51280000000286720000002752512000001468006400000469762048000090194313216009620726743040439804651110429509034655744
6256000000006553600000007340032000000469762048000001879048192000004810363371520007696581394432007036874417766402814749767106562478758911082496
751200000000014745600000000188743680000000140928614400000067645734912000002164663517184000046179488366592000633318697598976005066549580791808018014398509481984208215748530929664

All 15 rows

Technical details
Node id
n-698c1d594cf76bdc · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-833b492313a93e6e

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:             400,             2560,               4096;
  1:            8000,            76800,             245760,              262144;
  2:          160000,          2048000,            9830400,            20971520,            16777216;
  3:         3200000,         51200000,          327680000,          1048576000,          1677721600,           1073741824;
  4:        64000000,       1228800000,         9830400000,         41943040000,        100663296000,         128849018880,           68719476736;
  5:      1280000000,      28672000000,       275251200000,       1468006400000,       4697620480000,        9019431321600,         9620726743040,         4398046511104;
  6:     25600000000,     655360000000,      7340032000000,      46976204800000,     187904819200000,      481036337152000,       769658139443200,       703687441776640,       281474976710656;
  7:    512000000000,   14745600000000,    188743680000000,    1409286144000000,    6764573491200000,    21646635171840000,     46179488366592000,     63331869759897600,     50665495807918080,     18014398509481984;
  8:  10240000000000,  327680000000000,   4718592000000000,   40265318400000000,  225485783040000000,   865865406873600000,   2308974418329600000,   4222124650659840000,   5066549580791808000,   3602879701896396800,   1152921504606846976;
  9: 204800000000000, 7208960000000000, 115343360000000000, 1107296256000000000, 7086696038400000000, 31748398252032000000, 101594874406502400000, 232216855786291200000, 371546969258065920000, 396316767208603648000, 253642731013506334720, 73786976294838206464;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (20+84*x+64*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 2560*x + 4096*x^2.
%t Mathematica
P[0, x_] := 400 + 2560*x + 4096*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((20 + 84*x + 64*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] = 400 + 2560*x + 4096*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((20 + 84*x + 64*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.