Real-rooted atlas

A197208 → D

nonnegative coefficients

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

Recurrence

\(\left(5\,n + 6\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(60 + 47\,n + 20\,n x + 12\,n^{2} + 30\,n^{2} x + n^{3} + 10\,n^{3} x\right)\,P_{n-1}(x) + \left(60\,x + 39\,n x - 27\,n x^{2} + 3\,n^{2} x - 15\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(10\,x^{2} + 10\,x^{3} + 6\,n x^{2} + 6\,n x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 120 + 60\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
112060180
260012001801980
3210010500630042019320
45880588008820023520840177240
5141122469607408804939207056015121567944
630240846720444528059270402116800181440252013550040
759400249480020956320488980803492720074844004158003960115239960
8108900653400082328400307359360384199200164656800228690008712005940968932800
91887601557270028030860015697281603139456320235459224065405340062290800169884085808077898400
10312312343543208502694206802155360204064660802448775929611903771880226738512015459444031231201201266910203360
1149686071050980234468234025791505740110535024600198963044280154749034440515830114807034047020355254900546546016380551432634480
1276440013912080059682823208753480736051583011480013264202952001547490344400825328183680196953316560198942744007651644009172800218404526333862960
131142400259896000141903216002705621318402125845321600751638167280012527302788000100218422304003826521578880676405329600520311792001559376000148512002856037032917848080
1416646404660992003181127040077195349504078849535564803717192390912085185658958400973550388096005575788586368015769907112960212287211136012724508160030296448002330496036720302182770918960
15237211280651808067747518720205500806784026715104881920163725428490624500272142610240786141938387520643207040498880272875714151040587732307402246165024203520293572581120564562656035581680465122460298442276592
Technical details
Node id
n-fafdfaf5597dbd80 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-01b9b2797a61b9d5

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    120,       60;
   2:    600,     1200,       180;
   3:   2100,    10500,      6300,        420;
   4:   5880,    58800,     88200,      23520,         840;
   5:  14112,   246960,    740880,     493920,       70560,        1512;
   6:  30240,   846720,   4445280,    5927040,     2116800,      181440,        2520;
   7:  59400,  2494800,  20956320,   48898080,    34927200,     7484400,      415800,       3960;
   8: 108900,  6534000,  82328400,  307359360,   384199200,   164656800,    22869000,     871200,      5940;
   9: 188760, 15572700, 280308600, 1569728160,  3139456320,  2354592240,   654053400,   62290800,   1698840,    8580;
  10: 312312, 34354320, 850269420, 6802155360, 20406466080, 24487759296, 11903771880, 2267385120, 154594440, 3123120, 12012;
%F Formula
The row polynomials satisfy (n^3+6*n^2+5*n)*P(n,x) = (n^3+12*n^2+47*n+60+(10*n^3+30*n^2+20*n)*x)*P(n-1,x) + ((3*n^2+39*n+60)*x-(15*n^2+27*n)*x^2)*d/dx P(n-1,x) + ((6*n+10)*x^2+(6*n+10)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 120 + 60*x.
%t Mathematica
P[1, x_] := 120 + 60*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 12*n^2 + 47*n + 60 + (10*n^3 + 30*n^2 + 20*n)*x)*P[n - 1, x] + ((3*n^2 + 39*n + 60)*x - (15*n^2 + 27*n)*x^2)*D[P[n - 1, x], x] + ((6*n + 10)*x^2 + (6*n + 10)*x^3)*D[P[n - 1, x], {x, 2}])/(n^3 + 6*n^2 + 5*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 120 + 60*x; for(n = 2, N, my(i = n); v[i] = ((n^3 + 12*n^2 + 47*n + 60 + (10*n^3 + 30*n^2 + 20*n)*x)*v[i - 1] + ((3*n^2 + 39*n + 60)*x - (15*n^2 + 27*n)*x^2)*deriv(v[i - 1], x) + ((6*n + 10)*x^2 + (6*n + 10)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^3 + 6*n^2 + 5*n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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