Real-rooted atlas

A116414 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-156\,n - 90\,n x + 54\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-390 - 576\,x - 270\,x^{2} - 390\,n - 693\,n x - 270\,n x^{2} + 81\,n x^{3} - 117\,n^{2} x + 135\,n^{2} x^{2} + 162\,n^{2} x^{3} + 135\,n^{3} x^{2} + 81\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-169 - 585\,x - 567\,x^{2} - 135\,x^{3} + 117\,n x^{2} + 270\,n x^{3} + 81\,n x^{4} + 117\,n^{2} x^{2} + 270\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15611
2226027109
3954144051081022
44092460380721604059241
5176013512288902570410125145881449
6757370740193833237600145800437405103704389
732585358602120082518862201589625739206178605174966003164
8140206177630070363351352160014584050928746034649376998405904950569777
9603275864693039572550900946801188573759672226249294980153264962657205196830421972583
102595757415254002156649305681988008883897758836354805746080062435443206483580298415006495393493489309
1111168960197260752114647130034333696806216122250732621380458178792703141826704113816947526473635035724645212576428731068954
12480575299287413205973035094200481372004129057822556358202680530073206463492551520016100005095508805550010503045631275458406908733234952407625
1320678076543402695183060509751011383271733626298503887540845845359844483667480034694148096019495913085078312752100219349320304068712296449067645223205221911953428805
1488973123820155808220154637595099631439565120161792614971028199637999003494765999655315508517280020974231372051026163363500364955828391917479742401544792698815624365407174453515492236233141
Technical details
Node id
n-ac3aa4393b6caca9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8eeeb5cf78239d9a

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:      5,       6;
  2:     22,      60,       27;
  3:     95,     414,      405,      108;
  4:    409,    2460,     3807,     2160,       405;
  5:   1760,   13512,    28890,    25704,     10125,     1458;
  6:   7573,   70740,   193833,   237600,    145800,    43740,     5103;
  7:  32585,  358602,  1200825,  1886220,   1589625,   739206,   178605,    17496;
  8: 140206, 1776300,  7036335, 13521600,  14584050,  9287460,  3464937,   699840,   59049;
  9: 603275, 8646930, 39572550, 90094680, 118857375, 96722262, 49294980, 15326496, 2657205, 196830;
%F Formula
The row polynomials satisfy -(156*n+90*n*x-54*n^3*x^2)*P(n,x) = -(390*n+390+(117*n^2+693*n+576)*x-(135*n^3+135*n^2-270*n-270)*x^2-(81*n^3+162*n^2+81*n)*x^3)*P(n-1,x) - (169+585*x-(117*n^2+117*n-567)*x^2-(270*n^2+270*n-135)*x^3-(81*n^2+81*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[(-(390*n + 390 + (117*n^2 + 693*n + 576)*x - (135*n^3 + 135*n^2 - 270*n - 270)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*P[n - 1, x] - (169 + 585*x - (117*n^2 + 117*n - 567)*x^2 - (270*n^2 + 270*n - 135)*x^3 - (81*n^2 + 81*n)*x^4)*D[P[n - 1, x], x])/(-156*n - 90*n*x + 54*n^3*x^2)]];
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] = (-(390*n + 390 + (117*n^2 + 693*n + 576)*x - (135*n^3 + 135*n^2 - 270*n - 270)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*v[i - 1] - (169 + 585*x - (117*n^2 + 117*n - 567)*x^2 - (270*n^2 + 270*n - 135)*x^3 - (81*n^2 + 81*n)*x^4)*deriv(v[i - 1], x))/(-156*n - 90*n*x + 54*n^3*x^2)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023