Real-rooted atlas

A204420 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 36\,x + 27\,x^{2} + 108\,n - 120\,n x - 72\,n x^{2} - 144\,n^{2} + 112\,n^{2} x + 36\,n^{2} x^{2} + 80\,n^{3} - 32\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(-36\,x^{2} + 12\,x^{3} + 120\,n x^{2} - 32\,n x^{3} - 112\,n^{2} x^{2} + 16\,n^{2} x^{3} + 32\,n^{3} x^{2}\right)\,P_{n-2}'(x) + \left(4 - 4\,x - 12\,n + 8\,n x + 8\,n^{2}\right)\,P_{n-1}(x) + \left(4\,x^{2} - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
29482784
3225138012152403060
4110257392081270268802625195720
58930256384420807975034776005906253402019459440
61080560258118079201135664145578793600131236875134719205093552779539840
7182614682251428144517802156726221651240625786403393577687547335087803244591358648640539813514240
841088303506253322893974400053346327734700336653501184001054326648375017980522560001696457763008302694400164189025136868879347200
91187451971330625988256404985850016688095660536300113415307557552003941239876323750777736265506200906325607187006166307347200226088287425344594250043915646981606400
10428670161650355625365785197182004600064468821465850551754657979389583520000175948928966435625038859511486395600052831478396847150447520417344000022989829375012565472907500007922221807517397011281544294400
11189043541287806830625164912101315127015850030159085323125065056752294852510945507713200928774725462690356250224577401774454729000344060256728710021503422657019371990400220327683466015125884768135501250020130365612857519799007228008340335786274857164800
1210000403334124981340062588977702175550014226400016807576260238941722044501337294127166951624070400573793091881867381899375149562464446010856240000252313079087212734591002844170661012397516800216184751226803534175109366416092934000003527206062020140506557431193913600534435702050254759491323909524724121600
13625025208382811333753906255661110419313125702550625001100523473738680740630131250910509168413740061586420000410858734268626161502359375114133091586504010898377500208588457470164431034375002601486105050177785440000225106806837847599039375135077698708331754375005512837875573841312501459256129228898000022579908411623062515495473018025003188337543309617461002240000
14455643376911069462306597656254194452218184612261204827500008366728581528254985623607281257165863105130731804467464800003379281228242831305833970781259917721135492560910412893000019400489665379009422921310625263207695121105743584576000025300052018080244371153687517350724463765638195250000843184722809452873224375283519554136746127200006270093182701391793758200204321138830000480280605022968752477677852387845999767715840000
Technical details
Node id
n-5bbcb263521f680c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c40d4ba195f863cb

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:                1,                4;
  2:                9,               48,                27;
  3:              225,             1380,              1215,               240;
  4:            11025,            73920,             81270,             26880,             2625;
  5:           893025,          6384420,           8079750,           3477600,           590625,           34020;
  6:        108056025,        811807920,        1135664145,         578793600,        131236875,        13471920,         509355;
  7:      18261468225,     142814451780,      215672622165,      124062578640,      33935776875,      4733508780,      324459135,       8648640;
  8:    4108830350625,   33228939744000,    53346327734700,    33665350118400,   10543266483750,   1798052256000,   169645776300,    8302694400,    164189025;
  9: 1187451971330625, 9882564049858500, 16688095660536300, 11341530755755200, 3941239876323750, 777736265506200, 90632560718700, 6166307347200, 226088287425, 3445942500;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(32*n^3-112*n^2+120*n-36)*x-(36*n^2-72*n+27)*x^2)*P(n-2,x) + ((32*n^3-112*n^2+120*n-36)*x^2+(16*n^2-32*n+12)*x^3)*d/dx P(n-2,x) + (8*n^2-12*n+4+(8*n-4)*x)*P(n-1,x) - (8*n-4)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x - (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + ((32*n^3 - 112*n^2 + 120*n - 36)*x^2 + (16*n^2 - 32*n + 12)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 12*n + 4 + (8*n - 4)*x)*P[n - 1, x] - (8*n - 4)*x^2*D[P[n - 1, x], x]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x - (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + ((32*n^3 - 112*n^2 + 120*n - 36)*x^2 + (16*n^2 - 32*n + 12)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 12*n + 4 + (8*n - 4)*x)*v[i - 1] - (8*n - 4)*x^2*deriv(v[i - 1], 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.