Real-rooted atlas

A059364 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
22815245
3280234656585
4364041621781338249945
5582408479249306143132074120208845
6110656019604881445566567783125284147247205221125
724344320508766564552586822610188673072912009161189085040151412625
86086080001466670960154515994892946164434914972983868732125800281077264403204996616625
917041024000465442588805646451718439931365564181413472085490672057110705937214338364410824336362880184874815125
105282717440001613282265280221584262150418025175043249616954190883516243058478922556110215515486684176939085611949264036288007579867420125
111796123929600060662586203520930847540492168565986398356052525336796556225338760087666901536441785150900771937823082964262823526049176143779824039916800341094033905625
12664565853952000246005056108224041720869342632324286432015982312297135582927529214640574538830145257633604512371386517229184062664878940977236404195310483355111249921873050048047900160016713607661375625
132658263415808000010704137854466560019886413466459840022569441078471449617457784937878173697189923935899356400632813185286621238099260260232128684239743301584920510435689566074589890330451108646440962626565702406227020800885821206052908125
141143053268797440000497493615563374080010049737090203049600124889575490471009281066656927427361759266232566205466166123083380031799323450109326862037130107129667612733262928861316130514087320950078826280745610702097643423848284633753766439449538777608717829120050491808745015763125
Technical details
Node id
n-26a40306b1f4a278 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-116058163b8395d0

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:           4,           1;
  2:          28,          15,           2;
  3:         280,         234,          65,           6;
  4:        3640,        4162,        1781,         338,          24;
  5:       58240,       84792,       49306,       14313,        2074,        120;
  6:     1106560,     1960488,     1445566,      567783,      125284,      14724,        720;
  7:    24344320,    50876656,    45525868,    22610188,     6730729,    1200916,     118908,      5040;
  8:   608608000,  1466670960,  1545159948,   929461644,   349149729,   83868732,   12580028,   1077264,    40320;
  9: 17041024000, 46544258880, 56464517184, 39931365564, 18141347208, 5490672057, 1107059372, 143383644, 10824336, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+n*x)*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[(3*n + 1 + n*x)*P[n - 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] = (3*n + 1 + n*x)*v[i - 1]); 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.