Real-rooted atlas

A075504 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(-9\,x^{2}\right)\,P_{n-2}'(x) + \left(9 + 2\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 81 + 54\,x + 3\,x^{2}\)
\(P_{1}(x) = 729 + 1134\,x + 162\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
081543138
1729113416242029
26561218706075360534871
359049406782196830210606756684402
4531441744017459245831020600567001134714974639
5478296913498601417112400244641044382725012927617648359492327
64304672124393141904822827075183524292022802755511573604261954259299380296620
738742048943993748862133875302310724991812201255529362589860768230005640486000364510264240049483
8348678440179266232049436806237856632788470927000655612189650635911671782938730949692841608419954950117987456036451
9313810596091427489533769410056583569364210529245175288432988137193000422065765519225932974482883470091761461462751381050653412257641182979896
10282429536481257010878197710273668990673487539246160440286801616081738376105267501278011968212618745218888973615119042122823359828667925021660218424138824363028006917
112541865828329462676066663174274276143799138190144935097460678980776294482819605751638436709869759816515829192262408890931402496027290186223304938805872052978325432727241064714319597630345934303
122287679245496183286775731028014201239602357667874353166983669832942003674494044514074600977911161268022982123164818859733651828899394605568032107348668681074771681985001067754067389313995604791150132301512196591720037838834
1320589113209464914992077167434141745445962280016686808219408433590950082952417199810496186247475057216595561890130548890027994297591349273756735709856170435332070610944250102969345640590018795650285412217073414376156958074068266801620016488932300919737760497
14185302018885184126986150683645644430147265862717962055947570596489231422523536407982520143170687400655329609388629630201929262747033122968770230263276598217354629536369166099622236616713193026731091350029934321803022784394827451851241880381215425504794009501300195841720532483142005091074083
Technical details
Node id
n-955637f4530b33f6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bbb5df5595f8ddbd

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          81,             54,               3;
  1:         729,           1134,             162,               4;
  2:        6561,          21870,            6075,             360,              5;
  3:       59049,         406782,          196830,           21060,            675,             6;
  4:      531441,        7440174,         5924583,         1020600,          56700,          1134,            7;
  5:     4782969,      134986014,       171124002,        44641044,        3827250,        129276,         1764,          8;
  6:    43046721,     2439314190,      4822827075,      1835242920,      228027555,      11573604,       261954,       2592,         9;
  7:   387420489,    43993748862,    133875302310,     72499181220,    12555293625,     898607682,     30005640,     486000,      3645,      10;
  8:  3486784401,   792662320494,   3680623785663,   2788470927000,   655612189650,   63591167178,   2938730949,   69284160,    841995,    4950,   11;
  9: 31381059609, 14274895337694, 100565835693642, 105292451752884, 32988137193000, 4220657655192, 259329744828, 8347009176, 146146275, 1381050, 6534, 12;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 9*x^2*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 81 + 54*x + 3*x^2, P(1,x) = 729 + 1134*x + 162*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 81 + 54*x + 3*x^2;
P[1, x_] := 729 + 1134*x + 162*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 9*x^2*D[P[n - 2, x], x] + (9 + 2*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 81 + 54*x + 3*x^2; v[2] = 729 + 1134*x + 162*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 9*x^2*deriv(v[i - 2], x) + (9 + 2*x)*v[i - 1] + 9*x*deriv(v[i - 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.