Real-rooted atlas

Veronese parent of A098172 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22Pn(1)
1112
22795124
34246185703491288
4868258575833819553253751313456
516176888272056418361910873804446196662013217141472
632432272010600286105669585305993859058564790362341567451401235205211497664
764102477283656012153230156057912588068510760101065350858143562263298376127100428191114721622942515971968
8128236820832115920457800136474831887385984767917350511612615122277611075265179059894853275247690510431653584359880521315346539929171663616

All 15 rows

Technical details
Node id
n-a9043bb05acfd8df · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9835034ff7571c00

Observed on the stored rows ?

  • Degrees: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43 (affine: slope 3, 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1,     1;
   2:   2,     7,      9,      5,       1;
   3:   4,    24,     61,     85,      70,       34,        9,         1;
   4:   8,    68,    258,    575,     833,      819,      553,       253,        75,        13,          1;
   5:  16,   176,    888,   2720,    5641,     8361,     9108,      7380,      4446,      1966,        620,        132,         17,          1;
   6:  32,   432,   2720,  10600,   28610,    56695,    85305,     99385,     90585,     64790,      36234,      15674,       5140,       1235,        205,        21,         1;
   7:  64,  1024,   7728,  36560,  121532,   301560,   579125,    880685,   1076010,   1065350,     858143,     562263,     298376,     127100,      42819,     11147,      2162,       294,        25,        1;
   8: 128,  2368,  20832, 115920,  457800,  1364748,  3188738,   5984767,   9173505,  11612615,   12227761,   10752651,    7905989,    4853275,    2476905,   1043165,    358435,     98805,     21315,     3465,      399,      29,       1;
   9: 256,  5376,  54016, 345600, 1580896,  5503008, 15148336,  33830016,  62390545,  96220561,  125197384,  138268072,  130098892,  104474860,   71596792,  41789944,  20693590,   8639350,   3013560,   867160,   202188,   37228,    5208,    520,    33,    1;
  10: 512, 12032, 135936, 983040, 5111232, 20346912, 64480896, 166994640, 360027522, 654862247, 1014889769, 1349795853, 1548548241, 1537713852, 1324231524, 989562180, 641198928, 359503218, 173788174, 72053110, 25433766, 7567596, 1872468, 378180, 60726, 7455, 657, 37, 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 + x.
%t Mathematica
P[1, x_] := 1 + x;
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 + x; 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.