Real-rooted atlas

Veronese parent of A098172 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 10\,x + 7\,x^{2} + x^{3}\right)\,P_{n-1}(x) + \left(10\,x + 14\,x^{2} + 3\,x^{3}\right)\,P_{n-1}'(x) + \left(7\,x^{2} + 3\,x^{3}\right)\,P_{n-1}''(x) + \left(x^{3}\right)\,P_{n-1}'''(x)\)
\(P_{1}(x) = 1 + 3\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23Pn(1)
11315
224611575161255
345725258115229191319951938130304
4869042030441158836225411619332408374591969532587618736916618379
51682928747533610170986442188239673584073264292456831232278089916277159164831758593120410492010912329955413
6329952962711024808466533200704993121762300987020163631616085243148064202921627246698225297181403811255806419159509327915210768497195334075101068615811216014303146
7641194387297835766086892611181121117107947793665311538383664175921122288192253282904813136214780823819264114636891066880403487806465389677219661382161729951171119094477521516017651471952159194395925391120070204402161887654671626505
812814332710435249541126455571486258496172403739839801617608995740899200783176350818831201797084645973122082377040066154661281956548615382105601058501035975252563917467819774624810214618932419720191714512595603226290819560245526589829890081199110540007211019636268544690801191304901992657844356942831868222074880332913

All 15 rows

Technical details
Node id
n-3537990b4764b63a · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-f25876d70728b9b9

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1,           3,               1;
   2:   2,          46,             115,                 75,                   16,                      1;
   3:   4,         572,            5258,              11522,                 9191,                   3199,                     519,                      38,                        1;
   4:   8,        6904,          203044,            1158836,              2254116,                1933240,                  837459,                  196953,                    25876,                     1873,                       69,                        1;
   5:  16,       82928,         7475336,          101709864,            421882396,              735840732,               642924568,               312322780,                 89916277,                 15916483,                  1758593,                   120410,                     4920,                     109,                       1;
   6:  32,      995296,       271102480,         8466533200,          70499312176,           230098702016,            363161608524,            314806420292,             162724669822,              52971814038,              11255806419,               1595093279,                152107684,                 9719533,                  407510,                  10686,                   158,                    1;
   7:  64,    11943872,      9783576608,       689261118112,       11171079477936,         65311538383664,         175921122288192,         253282904813136,          214780823819264,          114636891066880,           40348780646538,            9677219661382,            1617299511711,            190944775215,             16017651471,              952159194,              39592539,              1120070,               20440,                216,                 1;
   8: 128,   143327104,    352495411264,     55571486258496,     1724037398398016,      17608995740899200,       78317635081883120,      179708464597312208,       237704006615466128,       195654861538210560,       105850103597525256,        39174678197746248,        10214618932419720,        1917145125956032,         262908195602455,         26589829890081,         1991105400072,         110196362685,          4469080119,          130490199,           2657844,            35694,            283,             1;
   9: 256,  1719926528,  12693274656896,   4461230132046464,   262500196381470656,    4614208237618743232,    33198584637879891840,   118311642406816506304,    236567575947086597552,    289264664254323418064,    230113394354733551008,    124682010396056208944,     47632188118438349064,    13172435169027028536,     2690188579591613328,     411829197501075864,     47758732408219721,     4223462710244215,     285616315250061,     14752126647822,      578494891530,      17018515383,      367984131,       5653056,       58203,       359,        1;
  10: 512, 20639120896, 456999165886720, 357456931847886080, 39687480580754317056, 1189626888286220701696, 13664826296276921642688, 74357820681120201551424, 220139239664519822531488, 390141652276439615228832, 443397716917361460252560, 340123915250999397378256, 183094130479256365909120, 71285229187120058167344, 20551352197576552033256, 4468866685271257044504, 743487419432489554650, 95666774011352559630, 9594473599758445699, 753664685439160971, 46465884273586240, 2246450228842125, 84800168966520, 2478239933964, 55285309296, 920691261, 11046945, 89965, 444, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+10*x+7*x^2+x^3)*P(n-1,x) + (10*x+14*x^2+3*x^3)*d/dx P(n-1,x) + (7*x^2+3*x^3)*d^2/dx^2 P(n-1,x) + x^3*d^3/dx^3 P(n-1,x) for n >= 2, with P(1,x) = 1 + 3*x + x^2.
%t Mathematica
P[1, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 10*x + 7*x^2 + x^3)*P[n - 1, x] + (10*x + 14*x^2 + 3*x^3)*D[P[n - 1, x], x] + (7*x^2 + 3*x^3)*D[P[n - 1, x], {x, 2}] + x^3*D[P[n - 1, x], {x, 3}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 3*x + x^2; for(n = 2, N, my(i = n); v[i] = (2 + 10*x + 7*x^2 + x^3)*v[i - 1] + (10*x + 14*x^2 + 3*x^3)*deriv(v[i - 1], x) + (7*x^2 + 3*x^3)*deriv(deriv(v[i - 1], x), x) + x^3*deriv(deriv(deriv(v[i - 1], x), x), x)); 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

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.