Real-rooted atlas

A196347 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3} + 6\,n^{2} x + 10\,n^{2} x^{2} + 4\,n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12810
286472144
34857612967682688
438461442073624576960061440
53840768003456006144004800001382401658880
6460801105920622080014745600172800009953280225792051609600
764512018063360121927680361267200564480000487710720221276160412876801816657920
81032192033030144026011238409248440320180633600002080899072014161674240528482304083607552071345111040
9185794560668860416060197437440249707888640585252864000842764124160764730408960428070666240135444234240185794560003091621478400
1037158912001486356480001504935936000713451110400019508428800000337105649664003823652044800028538044416000135444234240003715891200000449622835200146495294668800
11817496064003596982681600404660551680002158189608960006744342528000001359659453644800185064758968320017265516871680001092583489536000449622835200000108808726118400117719433216007534043725824000
1219619905536009417554657280011654223888384006906206748672000242796331008000005594027466424320088831084304793600994493771808768007866601124659200043163792179200000156684565610496003390319676620800331576403558400417888291992371200
1351011754393600265261122846720035810251584307200233429788105113600911835109785600000236347660456427520042892723564314624005602314914522726400531782236026961920036473404391424000001765312772544921600572964025348915200112072824402739200999830386114560024864353373546086400
141428329123020800799864308891648001169801551754035200831858881247313920035743936303595520000102942536554355097600210174345465141657600313729635213272678400347431060870948454400285951490428764160000173000651709402316800748672993122582528002196627358293688320039193351135690752003213740526796800001579617743731163136000
Technical details
Node id
n-560b152c9c95876b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         2,          8;
  2:         8,         64,          72;
  3:        48,        576,        1296,          768;
  4:       384,       6144,       20736,        24576,         9600;
  5:      3840,      76800,      345600,       614400,       480000,       138240;
  6:     46080,    1105920,     6220800,     14745600,     17280000,      9953280,      2257920;
  7:    645120,   18063360,   121927680,    361267200,    564480000,    487710720,    221276160,     41287680;
  8:  10321920,  330301440,  2601123840,   9248440320,  18063360000,  20808990720,  14161674240,   5284823040,    836075520;
  9: 185794560, 6688604160, 60197437440, 249707888640, 585252864000, 842764124160, 764730408960, 428070666240, 135444234240, 18579456000;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (2*n+(6*n^2+6*n)*x+(2*n^3+10*n^2+6*n)*x^2+(2*n^3+4*n^2+2*n)*x^3)*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 + (6*n^2 + 6*n)*x + (2*n^3 + 10*n^2 + 6*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*x^2)]];
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 + (6*n^2 + 6*n)*x + (2*n^3 + 10*n^2 + 6*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*x^2)); 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

Same fingerprint ?

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.