Real-rooted atlas

A088996 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 36\,n x + 81\,n x^{2} + 12\,n^{2} + 60\,n^{2} x - 162\,n^{2} x^{2} - 4\,n^{3} - 24\,n^{3} x + 81\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(36\,n x^{2} + 36\,n x^{3} - 60\,n^{2} x^{2} - 72\,n^{2} x^{3} + 24\,n^{3} x^{2} + 36\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(1 - 6\,n - 12\,n x + 4\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(12\,n x^{2} - 12\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111213
2360162225
315408202525925040
41053576251916134448600138816
5945384843474091223856187515010497604535604
610395491364538010124579504550476006015124825719120171379332
71351357260912931582625203906561522133775244660564820536717327054387207348794840
8202702512188664017894631781178061235242346656225893040279841107275271487495790284821427701120352457804520
9344594252290991580378254874062861646425281216887739425316460646975651642909899285179124489472292572300542471425670400018691204978944
1065472907547663971020873326192364745448753635236534834667575112697073268524227342959657002300759257150208252233620062696121930988688000259275183552001085802384278016
11137493105751087367629320218788998878792077793748820801151603541612225410277388204216898361790271082451611488992659494417858304370730772128367352835208005413182263771040101824435722240068562672044312448
123162341432252698935619716059129147539613761791752151120963817447129419817515406292688414760842726803784355586783314362093326323211449756728072898921088947327997901600683671381016899968255336453731727360430208240926464004675398488882478720
1379058535806257240664312721001715060289120421519547558933860123213309052720258477255995451479985859708188598746481890655334259310118572985516869747840234685037142823521415202985348006847920893364347570438114803705201532928127624828699954310401945864966652006400342395032980921304320
14213458046676875208779770458917005314905558177272556558419261352147984487625278950435944252423703408117414825248519565917736297493652184612569858220541632414728122124044861660858400624998835808840006034074473269945382544445192825897164586060822233292367921021986566741996498735828019209381848946357888000026799506861085782436096
Technical details
Node id
n-d357aef3250c0f5a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-0b605401778b8fa3

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,         12;
  2:        3,         60,         162;
  3:       15,        408,        2025,         2592;
  4:      105,       3576,       25191,        61344,         48600;
  5:      945,      38484,      347409,      1223856,       1875150,       1049760;
  6:    10395,     491364,     5380101,     24579504,      55047600,      60151248,      25719120;
  7:   135135,    7260912,    93158262,    520390656,    1522133775,    2446605648,    2053671732,     705438720;
  8:  2027025,  121886640,  1789463178,  11780612352,   42346656225,   89304027984,  110727527148,   74957902848,   21427701120;
  9: 34459425, 2290991580, 37825487406, 286164642528, 1216887739425, 3164606469756, 5164290989928, 5179124489472, 2925723005424, 714256704000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(24*n^3-60*n^2+36*n)*x-(81*n^3-162*n^2+81*n)*x^2)*P(n-2,x) + ((24*n^3-60*n^2+36*n)*x^2+(36*n^3-72*n^2+36*n)*x^3)*d/dx P(n-2,x) + (4*n^2-6*n+1+(12*n^2-12*n)*x)*P(n-1,x) - (12*n^2-12*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (24*n^3 - 60*n^2 + 36*n)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*P[n - 2, x] + ((24*n^3 - 60*n^2 + 36*n)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 6*n + 1 + (12*n^2 - 12*n)*x)*P[n - 1, x] - (12*n^2 - 12*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];
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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (24*n^3 - 60*n^2 + 36*n)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*v[i - 2] + ((24*n^3 - 60*n^2 + 36*n)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 6*n + 1 + (12*n^2 - 12*n)*x)*v[i - 1] - (12*n^2 - 12*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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. [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.