Real-rooted atlas

Veronese parent of A098172 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 5\,x + 4\,x^{2} + x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1 + 2\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23Pn(1)
11214
22916146148
34288514615510443101576
4876326833140816521372806328881416912
516192106436088361140021746916488118266412258675214918182944
6324643152133203921085305142000184690189970155375101024519082081463751440226221995328
7641088875244288158092423092880685145981019566952141360192349314204068606394254761699195396613309245631926111943936
8128249623200136752573720182254845534869173505151582722078612023840376229804121865864012759264733018035200701401600457240120120247803864428301143327232

All 15 rows

Technical details
Node id
n-256337cf537e7b19 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-37798f32d2e092fd

Observed on the stored rows ?

  • Degrees: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44 (affine: slope 3, intercept -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1,     2,      1;
   2:   2,     9,     16,      14,       6,        1;
   3:   4,    28,     85,     146,     155,      104,       43,        10,         1;
   4:   8,    76,    326,     833,    1408,     1652,     1372,       806,       328,         88,         14,          1;
   5:  16,   192,   1064,    3608,    8361,    14002,    17469,     16488,     11826,       6412,       2586,        752,        149,         18,          1;
   6:  32,   464,   3152,   13320,   39210,    85305,   142000,    184690,    189970,     155375,     101024,      51908,      20814,       6375,       1440,        226,         22,          1;
   7:  64,  1088,   8752,   44288,  158092,   423092,   880685,   1459810,   1956695,    2141360,    1923493,    1420406,     860639,     425476,     169919,      53966,      13309,       2456,       319,        26,        1;
   8: 128,  2496,  23200,  136752,  573720,  1822548,  4553486,   9173505,  15158272,   20786120,   23840376,   22980412,   18658640,   12759264,    7330180,    3520070,    1401600,     457240,    120120,     24780,     3864,      428,      30,       1;
   9: 256,  5632,  59392,  399616, 1926496,  7083904, 20651344,  48978352,  96220561,  158611106,  221417945,  263465456,  268366964,  234573752,  176071652,  113386736,   62483534,   29332940,  11652910,   3880720,  1069348,   239416,   42436,    5728,    553,    34,    1;
  10: 512, 12544, 147968, 1118976, 6094272, 25458144, 84827808, 231475536, 527022162, 1014889769, 1669752016, 2364685622, 2898344094, 3086262093, 2861945376, 2313793704, 1630761108, 1000702146, 533291392, 245841284, 97486876, 33001362, 9440064, 2250648, 438906, 68181, 8112, 694, 38, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+5*x+4*x^2+x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 2*x + x^2.
%t Mathematica
P[1, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 5*x + 4*x^2 + x^3)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 2*x + x^2; for(n = 2, N, my(i = n); v[i] = (2 + 5*x + 4*x^2 + x^3)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.