Real-rooted atlas

A078812 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-248 - 192\,x + 128\,x^{2} + 504\,n + 192\,n x - 128\,n x^{2} - 384\,n^{2} + 128\,n^{3}\right)\,P_{n}(x) = \left(100\,n + 144\,n x - 64\,n x^{2} - 256\,n x^{3} + 192\,n^{3} + 256\,n^{3} x\right)\,P_{n-1}(x) + \left(35 + 128\,x - 160\,x^{2} - 512\,x^{3} - 256\,x^{4} - 80\,n - 384\,n x - 256\,n x^{2} + 80\,n^{2} + 384\,n^{2} x + 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
27411
3325616104
412138833664909
54151992312017922567575
61344868820256208648960102461136
7419334248107376163840125440430084096482201
81274312609249972810230401155840703488200704163843738019
937984442024212484854942728309760744038437560329175046553628588344
101115451493516845616026521216506042886053376044843008193331204128768262144216287525
11323711490382431995168118243072273521920412274688407314432257163264967311361835008010485761621869871
12930432157368001163402884957836801351278080246752665630681907202579595264141852672047316992080740352419430412072013216
132653057495645604096783681979687680622292608013384544256201475276802129618534415574302720758382592022722641923523215361677721689292258609
147514311153695764140501884875968753922708721792067185957888119067574272152246943744139843731456904580300803951663513610745806848152672665667108864656908837179
152116112047036597647131597602820670617611257918156831676138496064721646796897383677952010813029089288777225994245088372326402015279185925016387584065766686722684354564810204846600
Technical details
Node id
n-b4a0a2fa43623102 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-861ee3c08734918c

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 -1)
  • 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:      1;
   2:      7,       4;
   3:     32,      56,      16;
   4:    121,     388,     336,       64;
   5:    415,    1992,    3120,     1792,      256;
   6:   1344,    8688,   20256,    20864,     8960,     1024;
   7:   4193,   34248,  107376,   163840,   125440,    43008,     4096;
   8:  12743,  126092,  499728,  1023040,  1155840,   703488,   200704,    16384;
   9:  37984,  442024, 2124848,  5494272,  8309760,  7440384,  3756032,   917504,   65536;
  10: 111545, 1493516, 8456160, 26521216, 50604288, 60533760, 44843008, 19333120, 4128768, 262144;
%F Formula
The row polynomials satisfy (128*n^3-384*n^2+504*n-248+(192*n-192)*x-(128*n-128)*x^2)*P(n,x) = (192*n^3+100*n+(256*n^3+144*n)*x-64*n*x^2-256*n*x^3)*P(n-1,x) + (80*n^2-80*n+35+(384*n^2-384*n+128)*x+(256*n^2-256*n-160)*x^2-512*x^3-256*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((192*n^3 + 100*n + (256*n^3 + 144*n)*x - 64*n*x^2 - 256*n*x^3)*P[n - 1, x] + (80*n^2 - 80*n + 35 + (384*n^2 - 384*n + 128)*x + (256*n^2 - 256*n - 160)*x^2 - 512*x^3 - 256*x^4)*D[P[n - 1, x], x])/(128*n^3 - 384*n^2 + 504*n - 248 + (192*n - 192)*x - (128*n - 128)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((192*n^3 + 100*n + (256*n^3 + 144*n)*x - 64*n*x^2 - 256*n*x^3)*v[i - 1] + (80*n^2 - 80*n + 35 + (384*n^2 - 384*n + 128)*x + (256*n^2 - 256*n - 160)*x^2 - 512*x^3 - 256*x^4)*deriv(v[i - 1], x))/(128*n^3 - 384*n^2 + 504*n - 248 + (192*n - 192)*x - (128*n - 128)*x^2)); 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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023