Real-rooted atlas

A225470 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
24026470
3440366968910
46160600420763041614560
510472011438847300932088032276640
6209440024972001174776281000362402400646086080
7481712006162440032014248881255213955201276806272128152152000
812524512001698576800955619248293154848539086246110720418432158722564260256000
936321084800517636296003111011179210412729088214965979228502812824355968129715239168512132067936000
10116227471360017290783168001099050836544395427554400896145715201342021968013494472329022080038476809472010244490309824000
11406796149760006284229051520041924935912640160380660770883927365112000648936831840740710924805856622464315110400110105602252802048166141463488000
121545825369088000246936626952960017188321457107206932963827546241813160064101763251432983392041125751779203706938385922368744012810486220803058176052838440966645658539520000
13633788401326080001043356677888896007541085051319872031862815984361024882054902832646416957195360110722336442419625602342359773811217125727224329036838553633510963208282726412247048192285763317199360000
1427886689658347520004717527062976358400352674875815852288015527856043382824964518297892150864649225275764114009613671785718374784149792678440204812220039526323274013544084483281850091521034659225621954150428098561638413145112591170560000
Technical details
Node id
n-92200b86dc1e86f0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-144038514c6e54f6

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:           5,           2;
  2:          40,          26,           4;
  3:         440,         366,          96,           8;
  4:        6160,        6004,        2076,         304,         16;
  5:      104720,      114388,       47300,        9320,        880,        32;
  6:     2094400,     2497200,     1174776,      281000,      36240,      2400,       64;
  7:    48171200,    61624400,    32014248,     8812552,    1395520,    127680,     6272,     128;
  8:  1252451200,  1698576800,   955619248,   293154848,   53908624,   6110720,   418432,   15872,   256;
  9: 36321084800, 51763629600, 31110111792, 10412729088, 2149659792, 285028128, 24355968, 1297152, 39168, 512;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+2+2*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 + 2 + 2*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 + 2 + 2*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.