Real-rooted atlas

A038330 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 5\,x + 4\,x^{2} + 3\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(16 + 144\,x + 272\,x^{2} + 144\,x^{3} + 48\,n x + 144\,n x^{2} + 96\,n x^{3} + 16\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 2048\,x + 2304\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0256204823044608
140964915211059265536229376
265536104857635389444194304163840010485760
31048576209715209437184016777216013107200037748736452984832
416777216402653184226492416053687091206291456000362387865682208358418790481920
52684354567516192768507343011841503238553602348810240002029372047369207336140817179869184755914244096
6429496729613743895347210823317585923848290697216751619276800086586540687365892695130112219902325555234789235097629686813949952
768719476736247390116249622265110462464923589767331842164663517184003117115464744962828493662453761583296743997445009649854054468719476736001143492092887040

All 15 rows

Technical details
Node id
n-9caddd71807c8fdc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:            256,            2048,             2304;
  1:           4096,           49152,           110592,             65536;
  2:          65536,         1048576,          3538944,           4194304,            1638400;
  3:        1048576,        20971520,         94371840,         167772160,          131072000,           37748736;
  4:       16777216,       402653184,       2264924160,        5368709120,         6291456000,         3623878656,          822083584;
  5:      268435456,      7516192768,      50734301184,      150323855360,       234881024000,       202937204736,        92073361408,        17179869184;
  6:     4294967296,    137438953472,    1082331758592,     3848290697216,      7516192768000,      8658654068736,      5892695130112,      2199023255552,       347892350976;
  7:    68719476736,   2473901162496,   22265110462464,    92358976733184,    216466351718400,    311711546474496,    282849366245376,    158329674399744,     50096498540544,     6871947673600;
  8:  1099511627776,  43980465111040,  445302209249280,  2111062325329920,   5772436045824000,   9974769487183872,  11313974649815040,   8444249301319680,   4007719883243520,  1099511627776000,   133040906960896;
  9: 17592186044416, 774056185954304, 8708132091985920, 46443371157258240, 145135534866432000, 292593238290726912, 398251907673489408, 371546969258065920, 235119566483619840, 96757023244288000, 23415199625117696, 2533274790395904;
%F Formula
The row polynomials satisfy (1+(3*n+5)*x+(n^2+4*n+4)*x^2)*P(n,x) = (16+(48*n+144)*x+(16*n^2+144*n+272)*x^2+(16*n^2+96*n+144)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 2048*x + 2304*x^2.
%t Mathematica
P[0, x_] := 256 + 2048*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + (48*n + 144)*x + (16*n^2 + 144*n + 272)*x^2 + (16*n^2 + 96*n + 144)*x^3)*P[n - 1, x])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 2048*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16 + (48*n + 144)*x + (16*n^2 + 144*n + 272)*x^2 + (16*n^2 + 96*n + 144)*x^3)*v[i - 1])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*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.