Real-rooted atlas

A038334 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 64 + 320\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
064320400784
151238409600800021952
2409640960153600256000160000614656
332768409600204800051200006400000320000017210368
42621443932160245760008192000015360000015360000064000000481890304
52097152367001602752512001146880000286720000043008000003584000000128000000013492928512
61677721633554432029360128001468006400045875200000917504000001146880000008192000000025600000000377801998336
7134217728301989888030198988800176160768000660602880000165150720000027525120000002949120000000184320000000051200000000010578455953408

All 15 rows

Technical details
Node id
n-e873007dd974fa35 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9629d1e9c04dbfbe

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:         64,          320,           400;
  1:        512,         3840,          9600,           8000;
  2:       4096,        40960,        153600,         256000,          160000;
  3:      32768,       409600,       2048000,        5120000,         6400000,         3200000;
  4:     262144,      3932160,      24576000,       81920000,       153600000,       153600000,        64000000;
  5:    2097152,     36700160,     275251200,     1146880000,      2867200000,      4300800000,      3584000000,       1280000000;
  6:   16777216,    335544320,    2936012800,    14680064000,     45875200000,     91750400000,    114688000000,      81920000000,      25600000000;
  7:  134217728,   3019898880,   30198988800,   176160768000,    660602880000,   1651507200000,   2752512000000,    2949120000000,    1843200000000,     512000000000;
  8: 1073741824,  26843545600,  301989888000,  2013265920000,   8808038400000,  26424115200000,  55050240000000,   78643200000000,   73728000000000,   40960000000000,  10240000000000;
  9: 8589934592, 236223201280, 2952790016000, 22145925120000, 110729625600000, 387553689600000, 968884224000000, 1730150400000000, 2162688000000000, 1802240000000000, 901120000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = (8+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 320*x + 400*x^2.
%t Mathematica
P[0, x_] := 64 + 320*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 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] = 64 + 320*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + 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

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.