Real-rooted atlas

A108426 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 2\,n^{2} + 12\,n^{3} + 4\,n^{3} x + 8\,n^{4} x\right)\,P_{n}(x) = \left(-6 - 10\,x - 4\,x^{2} + 45\,n + 76\,n x + 31\,n x^{2} + 2\,n x^{3} - 108\,n^{2} - 174\,n^{2} x - 66\,n^{2} x^{2} - 2\,n^{2} x^{3} + 81\,n^{3} + 90\,n^{3} x + 15\,n^{3} x^{2} - 2\,n^{3} x^{3} + 54\,n^{4} x + 48\,n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-3\,x - 5\,x^{2} - 2\,x^{3} + 18\,n x + 35\,n x^{2} + 19\,n x^{3} + 2\,n x^{4} - 27\,n^{2} x - 48\,n^{2} x^{2} - 21\,n^{2} x^{3} - 18\,n^{3} x^{2} - 20\,n^{3} x^{3} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
2310619
312566320151
455330540336701331
5273200242903960165025212427
61428123763276040040250257722924120275
77752775202441883712803094001441443503534321192751
84326349031417907123255840339150021162967796881555841287012036067
924667531245501297890027457584343219802686068013294680403104068023848620123044947
1014307152003001093243150224967600328077750308897820192203088784502402014551029393001847561270569939
11841464012902448066528247518027009003004501500330702372024802677901274816400441282600980628001258472670543213224667463
12500671088344518004720613040141926928002661129900033530236740294441147001826736912079722893252390280750467432680534888002704156138537040019
1330083057254149502963334469392811014763760022947424500032572229976032785120368023843723904012573838777547571273750125588162702190366360225990180104006001458988334811
141822766520352401527202346478461608447322461761935844730820305659694340034696505844002895309331200179147264868081962800920027320933640064350959400101197073259500948104011660015433615473811
Technical details
Node id
n-1e24992b6bf1f3cd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-51b62033f048530c

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,       2;
  2:      3,      10,        6;
  3:     12,      56,       63,       20;
  4:     55,     330,      540,      336,       70;
  5:    273,    2002,     4290,     3960,     1650,      252;
  6:   1428,   12376,    32760,    40040,    25025,     7722,      924;
  7:   7752,   77520,   244188,   371280,   309400,   144144,    35035,    3432;
  8:  43263,  490314,  1790712,  3255840,  3391500,  2116296,   779688,  155584,  12870;
  9: 246675, 3124550, 12978900, 27457584, 34321980, 26860680, 13294680, 4031040, 680238, 48620;
%F Formula
The row polynomials satisfy (12*n^3+2*n^2-2*n+(8*n^4+4*n^3)*x)*P(n,x) = (81*n^3-108*n^2+45*n-6+(54*n^4+90*n^3-174*n^2+76*n-10)*x+(48*n^4+15*n^3-66*n^2+31*n-4)*x^2+(2*n^4-2*n^3-2*n^2+2*n)*x^3)*P(n-1,x) - ((27*n^2-18*n+3)*x+(18*n^3+48*n^2-35*n+5)*x^2+(20*n^3+21*n^2-19*n+2)*x^3+(2*n^3-2*n)*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[((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 90*n^3 - 174*n^2 + 76*n - 10)*x + (48*n^4 + 15*n^3 - 66*n^2 + 31*n - 4)*x^2 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^3)*P[n - 1, x] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 48*n^2 - 35*n + 5)*x^2 + (20*n^3 + 21*n^2 - 19*n + 2)*x^3 + (2*n^3 - 2*n)*x^4)*D[P[n - 1, x], x])/(12*n^3 + 2*n^2 - 2*n + (8*n^4 + 4*n^3)*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] = ((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 90*n^3 - 174*n^2 + 76*n - 10)*x + (48*n^4 + 15*n^3 - 66*n^2 + 31*n - 4)*x^2 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^3)*v[i - 1] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 48*n^2 - 35*n + 5)*x^2 + (20*n^3 + 21*n^2 - 19*n + 2)*x^3 + (2*n^3 - 2*n)*x^4)*deriv(v[i - 1], x))/(12*n^3 + 2*n^2 - 2*n + (8*n^4 + 4*n^3)*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. [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.