Real-rooted atlas

A178821 → basis A013610 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-120 + 36\,x - 70\,n + 21\,n x - 10\,n^{2} + 3\,n^{2} x\right)\,P_{n-2}(x) + \left(-28 - 12\,x + 21\,n + 9\,n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 25 + 15\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1251540
23755851351095
3437511655661594523590
443750167370164430548105670436030
5393750197278228633501615950377622306187254072
63281250203571903984309033005070128085302296350153090111744570
72578125019082349047475153053106273029929581087808050127502107217101622994780
81933593751663017345504839263572163114655463865935230526202554164335566036465324769522501136295
91396484375136859129554920533689586420820915834836037754702842787515760582695308206255532368693514073345300400992320
109775390625107530321899447692390145938283521175111439420992979803966141935500513148198349225975164517403051517106591591080493887097807593
11666503906258131375611453852332474445941977191928513377009583665117568546436256621853832685242250756520557084941686583369216385685116022524180565548991429569810
12444335937500595410901722031654320728940887094592511401473873743323801548073835217001067296925331004924570982058015255285924060312366336282040462146270029983901220967222620603746907015980
132905273437500424170116844602502016707619807921235626630201513207868121180186027571762830015303478885695008618952410523003355575542890208997355848222016274843279100189344988126012774778758037944887407297205182126160
14186767578125002950928235725401913245766801460676234994306382014643599709381300207197503569063001993485557238210013370190155570700633637428479010021284837060045405024671024005008135221237614085880121582605317704934201463588514086715572809598820
Technical details
Node id
n-96a7112290a52677 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-0fee5d4c7376c95e

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:         25,          15;
  2:        375,         585,         135;
  3:       4375,       11655,        6615,         945;
  4:      43750,      167370,      164430,       54810,        5670;
  5:     393750,     1972782,     2863350,     1615950,      377622,       30618;
  6:    3281250,    20357190,    39843090,    33005070,    12808530,     2296350,      153090;
  7:   25781250,   190823490,   474751530,   531062730,   299295810,    87808050,    12750210,     721710;
  8:  193359375,  1663017345,  5048392635,  7216311465,  5463865935,  2305262025,   541643355,   66036465,   3247695;
  9: 1396484375, 13685912955, 49205336895, 86420820915, 83483603775, 47028427875, 15760582695, 3082062555, 323686935, 14073345;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(10*n^2+70*n+120-(3*n^2+21*n+36)*x)*P(n-2,x) + (7*n^2+21*n-28+(3*n^2+9*n-12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 25 + 15*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 25 + 15*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(10*n^2 + 70*n + 120 - (3*n^2 + 21*n + 36)*x)*P[n - 2, x] + (7*n^2 + 21*n - 28 + (3*n^2 + 9*n - 12)*x)*P[n - 1, x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 25 + 15*x; for(n = 2, N, my(i = n + 1); v[i] = (-(10*n^2 + 70*n + 120 - (3*n^2 + 21*n + 36)*x)*v[i - 2] + (7*n^2 + 21*n - 28 + (3*n^2 + 9*n - 12)*x)*v[i - 1])/(n^2 - n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

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.