Real-rooted atlas

A059364 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + 2\,n + 5\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
215412884
31054376062801428
494552981113510398364031416
5103957339820725329253820640858240848232
613513511516794088851774080182415264678872110656027143424
720270252024815586669703206066737293920512251496652119533584243443201004306688
8344594253948942601979588826566987710410148317129116234558848319487228340219304060860800042180880896
9654729075846785484048669226974163156152104351574584363504998541408483527022084297587310144106824957120170410240001982501402112
10137493105751981215529651284557266494493502523037812440906986847215037814848212580902224741221238671197628114685307139843669435174720528271744000103090072909824
1131623414322550242722777453628094993017215718052735949045393171853033391757781170368113247360821743901365996193957452985891027140984474327053294016136911046052480179612392960005876134155859968
12790585358062513730747024295010929218225308655271908331403614171639724755594553973491887821434667068781087395947831651398893887307518913485495032448336144336566808209728771231905925514739686341760664565853952000364320317663318016
132134580466768754023535839784650350011880180513551860583978491321786743035900962498451759401708734165542340025384448826440949282100205093335905356712236659207948431264129128531677624997134557888183049878130564588512002385328655103744002658263415808000024409461283442307072
146190283353629375125861235360860475118804649363422924569007446224010414272755531522030392915980017703927429544053174261634965728786167289129005907766043697367257685745018374862355405223623620128268257935265743144352184136134257236225871872493934145275314592001102780960753041920011430532687974400001757481212407846109184
Technical details
Node id
n-fc85eff121403457 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5ef78deb29f8c984

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:         1;
  1:         3,          4;
  2:        15,         41,          28;
  3:       105,        437,         606,          280;
  4:       945,       5298,       11135,        10398,         3640;
  5:     10395,      73398,      207253,       292538,       206408,        58240;
  6:    135135,    1151679,     4088851,      7740801,      8241526,      4678872,      1106560;
  7:   2027025,   20248155,    86669703,    206066737,    293920512,    251496652,    119533584,     24344320;
  8:  34459425,  394894260,  1979588826,   5669877104,  10148317129,  11623455884,   8319487228,   3402193040,    608608000;
  9: 654729075, 8467854840, 48669226974, 163156152104, 351574584363, 504998541408, 483527022084, 297587310144, 106824957120, 17041024000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+1+(5*n+2)*x+(3*n+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (5*n + 2)*x + (3*n + 1)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 + (5*n + 2)*x + (3*n + 1)*x^2)*v[i - 1])/(1 + x)); 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

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.