Real-rooted atlas

A090452 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,x + 2\,n + 6\,n x + 3\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(12\,x + 24\,x^{2} + 12\,x^{3} + 2\,n + 14\,n x + 36\,n x^{2} + 24\,n x^{3} + 3\,n^{2} + 15\,n^{2} x + 12\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4} + 3\,n x + 14\,n x^{2} + 19\,n x^{3} + 8\,n x^{4} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
116613
2114486025148
3124153420570378981644
41363541584385554604508201637818192
51506904440161753649853060499202943099001452201616
6166120610400518251648983532485226485366253762001720624633255772241088
71841953215601391255831281685257347594452082105694150450265225105089369362102102145024991168
811042988408803291401739304642938817290304346474355241060060082792519398883333930615415400485628093350482654279539968
9112643747238470776045722882085518470056864178145163348956850531794142632681712585946998418256748225735120891072002427773440814283191243135571200
1011506180121380141187510897488597584402433391207580129851843090050354463124454341001006660816435651738906050567868003071397888142927551049203882011807588017635800123450635260019712
111176848119470026504502400967815518995874917972827870702758167179900191432235393626306564455864953144701765872507182169830059612560128397675161182100965644885902657502621403400561985116755083564784276397398690816
1212041135830096047313754958830837195281220923626249118525185315078555008777922179419954972425637323760628757733503282074028004380078655402636869027877096249712441882429071619522013585535040049525274106135699302642628617928320932800185725004487919046656
13123414898450840809503597013202833828996539102678427108254745108603207570352866820878942138283320208700727662738615857251305993843492520782070437990485774243878187889308265124605634325010038502891741602003740620882839275376940275988479208686083008001212041350013559410807220988050774714290176
14126619194657384133556151812244981766220092129772500007437541211134086125073012725625652743924728484768101053633352492189320991025040133765983680624789929107688275362555996993265882144344892938995928987236317282194049346628722391280716228541321315729005463649924939318801495144753375340115220900552225057305700568860281150505575379878461440
Technical details
Node id
n-d4944ce4c8038cb8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9c33180785a5ceeb

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,   6,    6;
  2: 1,  14,   48,    60,     25;
  3: 1,  24,  153,   420,    570,     378,       98;
  4: 1,  36,  354,  1584,   3855,    5460,     4508,     2016,       378;
  5: 1,  50,  690,  4440,  16175,   36498,    53060,    49920,     29430,      9900,      1452;
  6: 1,  66, 1206, 10400,  51825,  164898,   353248,   522648,    536625,    376200,    172062,     46332,      5577;
  7: 1,  84, 1953, 21560, 139125,  583128,  1685257,  3475944,   5208210,   5694150,   4502652,   2510508,    936936,    210210,     21450;
  8: 1, 104, 2988, 40880, 329140, 1739304,  6429388, 17290304,  34647435,  52410600,  60082792,  51939888,  33339306,  15415400,   4856280,   933504,    82654;
  9: 1, 126, 4374, 72384, 707760, 4572288, 20855184, 70056864, 178145163, 348956850, 531794142, 632681712, 585946998, 418256748, 225735120, 89107200, 24277734, 4081428, 319124;
%F Formula
The row polynomials satisfy (n^3+3*n^2+2*n+(4*n^2+6*n-4)*x)*P(n,x) = (n^3+3*n^2+2*n+(n^3+15*n^2+14*n+12)*x+(12*n^2+36*n+24)*x^2+(24*n+12)*x^3)*P(n-1,x) + ((2*n^2+3*n+4)*x+(4*n^2+14*n+12)*x^2+(2*n^2+19*n+12)*x^3+(8*n+4)*x^4)*d/dx 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[((n^3 + 3*n^2 + 2*n + (n^3 + 15*n^2 + 14*n + 12)*x + (12*n^2 + 36*n + 24)*x^2 + (24*n + 12)*x^3)*P[n - 1, x] + ((2*n^2 + 3*n + 4)*x + (4*n^2 + 14*n + 12)*x^2 + (2*n^2 + 19*n + 12)*x^3 + (8*n + 4)*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + (4*n^2 + 6*n - 4)*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] = ((n^3 + 3*n^2 + 2*n + (n^3 + 15*n^2 + 14*n + 12)*x + (12*n^2 + 36*n + 24)*x^2 + (24*n + 12)*x^3)*v[i - 1] + ((2*n^2 + 3*n + 4)*x + (4*n^2 + 14*n + 12)*x^2 + (2*n^2 + 19*n + 12)*x^3 + (8*n + 4)*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + (4*n^2 + 6*n - 4)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.