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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21Pn(1)
111
2254112
342041442681144
4860198377456363190631211728
51616072819923649471243962984146250411616120736
632400232082802033036365489405044540140246501158440901050185201248832
764960676829792917402098203693055113805646305007203574232048409353633564925518922702412985984
8128224018592973283604721004276218446238003055373200623941559883464764305314168417115917653142778518058418221309437128135831808

All 15 rows

Technical details
Node id
n-73e4b84e56cef63a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f9a4247dd793824e

Observed on the stored rows ?

  • Degrees: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42 (affine: slope 3, intercept -3)
  • 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:   2,     5,      4,      1;
   3:   4,    20,     41,     44,      26,        8,        1;
   4:   8,    60,    198,    377,     456,      363,      190,        63,        12,         1;
   5:  16,   160,    728,   1992,    3649,     4712,     4396,      2984,      1462,       504,       116,        16,         1;
   6:  32,   400,   2320,   8280,   20330,    36365,    48940,     50445,     40140,     24650,     11584,      4090,      1050,       185,        20,         1;
   7:  64,   960,   6768,  29792,   91740,   209820,   369305,    511380,    564630,    500720,    357423,    204840,     93536,     33564,      9255,      1892,       270,        24,        1;
   8: 128,  2240,  18592,  97328,  360472,  1004276,  2184462,   3800305,   5373200,   6239415,   5988346,   4764305,   3141684,   1711591,    765314,    277851,     80584,     18221,     3094,      371,      28,       1;
   9: 256,  5120,  48896, 296704, 1284192,  4218816, 10929520,  22900496,  39490049,  56730512,  68466872,  69801200,  60297692,  44177168,  27419624,  14370320,   6323270,   2316080,   697480,   169680,   32508,    4720,    488,    32,    1;
  10: 512, 11520, 124416, 858624, 4252608, 16094304, 48386592, 118608048, 241419474, 413442773, 601446996, 748348857, 800199384, 737514468, 586717056, 402845124, 238353804, 121149414, 52638760, 19414350, 6019416, 1548180, 324288, 53892, 6834, 621, 36, 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.
%t Mathematica
P[1, x_] := 1;
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; 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.