Real-rooted atlas

A033282 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 10\,x - 4\,x^{2} + 5\,n + 5\,n x - 2\,n x^{2} + 6\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(22 + 46\,x + 12\,x^{2} - 16\,x^{3} + 47\,n + 71\,n x + 6\,n x^{2} - 8\,n x^{3} + 30\,n^{2} + 40\,n^{2} x + 6\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(4 + 14\,x + 10\,x^{2} - 8\,x^{3} - 8\,x^{4} + 8\,n + 24\,n x + 16\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2417020131
32316905881121621
4128955728288470467220525
57183408408784086880369604224263927
6400812831228008201144440847440288288274563431647
72241431894270665742012507880133733607911904224224018304044994257
81256465123695445202243212143331217063446414568153671751680174254081244672593819513
9705973579369164388937952108647011218964895042127250944151125811263749836813544294485995527878376387
103975327350261523028133905809158014320191295686402665018137624866849280151244620805578959360105344512060196864104977436123
11224298231315060484219837582380737699793601795663161602993326533123468346506242769907430401472681548804826167808082006650884260085761403863334573
1212678548731959009678013706640288057343014432015946594646403097608408576429421605888042579242956802971804239360140331789312041371518566463901286400304291840018831544249573
137178461215121011665424931553019072433127232595213551657497664300743865776644858033977139257625567853568499271134602243094261562163213141147402240352065658880049843003392021909012480253274839291247
14407047787857434696367146245284042452319523950047121111153186983842774476669124165120148201168007076438526094087333780103569925641310734259203143812825753601213054095052802978119452672038915883417601588403404803414230910872183
Technical details
Node id
n-d6de9b1f4cdfd204 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e8c7cc224f2c532c

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:       7,        4;
  2:      41,       70,        20;
  3:     231,      690,       588,        112;
  4:    1289,     5572,      8288,       4704,        672;
  5:    7183,    40840,     87840,      86880,      36960,       4224;
  6:   40081,   283122,    800820,    1144440,     847440,     288288,      27456;
  7:  224143,  1894270,   6657420,   12507880,   13373360,    7911904,    2242240,    183040;
  8: 1256465, 12369544,  52022432,  121433312,  170634464,  145681536,   71751680,  17425408,   1244672;
  9: 7059735, 79369164, 388937952, 1086470112, 1896489504, 2127250944, 1511258112, 637498368, 135442944, 8599552;
%F Formula
The row polynomials satisfy (2*n^3+6*n^2+5*n+2+(5*n+10)*x-(2*n+4)*x^2)*P(n,x) = (6*n^3+30*n^2+47*n+22+(8*n^3+40*n^2+71*n+46)*x+(6*n+12)*x^2-(8*n+16)*x^3)*P(n-1,x) + (4*n^2+8*n+4+(12*n^2+24*n+14)*x+(8*n^2+16*n+10)*x^2-8*x^3-8*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[((6*n^3 + 30*n^2 + 47*n + 22 + (8*n^3 + 40*n^2 + 71*n + 46)*x + (6*n + 12)*x^2 - (8*n + 16)*x^3)*P[n - 1, x] + (4*n^2 + 8*n + 4 + (12*n^2 + 24*n + 14)*x + (8*n^2 + 16*n + 10)*x^2 - 8*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 6*n^2 + 5*n + 2 + (5*n + 10)*x - (2*n + 4)*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] = ((6*n^3 + 30*n^2 + 47*n + 22 + (8*n^3 + 40*n^2 + 71*n + 46)*x + (6*n + 12)*x^2 - (8*n + 16)*x^3)*v[i - 1] + (4*n^2 + 8*n + 4 + (12*n^2 + 24*n + 14)*x + (8*n^2 + 16*n + 10)*x^2 - 8*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 6*n^2 + 5*n + 2 + (5*n + 10)*x - (2*n + 4)*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