Real-rooted atlas

A104684 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-15 + 34\,x - 8\,x^{2} + 39\,n - 50\,n x + 8\,n x^{2} - 32\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(-17 + 242\,x + 432\,x^{2} - 160\,x^{3} + 197\,n + 30\,n x - 576\,n x^{2} + 96\,n x^{3} - 208\,n^{2} - 272\,n^{2} x + 192\,n^{2} x^{2} + 56\,n^{3} + 96\,n^{3} x\right)\,P_{n-1}(x) + \left(-1 + 12\,x + 92\,x^{2} + 96\,x^{3} - 64\,x^{4} + 12\,n + 56\,n x - 128\,n x^{3} - 8\,n^{2} - 48\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
27411
3456416125
4291708432641495
5185765287104256025618305
611703541808736057600140801024225947
7730214206729079209036804108807372840962793997
84520353120292847492811439680796544026880003727361638434529495

All 15 rows

Technical details
Node id
n-bb87b1209ecaa616 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-56becc4b901ca19b

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:       45,        64,        16;
   4:      291,       708,       432,         64;
   5:     1857,      6528,      7104,       2560,        256;
   6:    11703,     54180,     87360,      57600,      14080,       1024;
   7:    73021,    420672,    907920,     903680,     410880,      73728,      4096;
   8:   452035,   3120292,   8474928,   11439680,    7965440,    2688000,    372736,    16384;
   9:  2780673,  22388224,  73459456,  125970432,  120243200,   62849024,  16515072,  1835008,   65536;
  10: 17017191, 156637764, 603217152, 1258521600, 1538546688, 1112444928, 456916992, 96731136, 8847360, 262144;
%F Formula
The row polynomials satisfy (8*n^3-32*n^2+39*n-15+(16*n^2-50*n+34)*x+(8*n-8)*x^2)*P(n,x) = (56*n^3-208*n^2+197*n-17+(96*n^3-272*n^2+30*n+242)*x+(192*n^2-576*n+432)*x^2+(96*n-160)*x^3)*P(n-1,x) - (8*n^2-12*n+1+(48*n^2-56*n-12)*x+(64*n^2-92)*x^2+(128*n-96)*x^3+64*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[((56*n^3 - 208*n^2 + 197*n - 17 + (96*n^3 - 272*n^2 + 30*n + 242)*x + (192*n^2 - 576*n + 432)*x^2 + (96*n - 160)*x^3)*P[n - 1, x] - (8*n^2 - 12*n + 1 + (48*n^2 - 56*n - 12)*x + (64*n^2 - 92)*x^2 + (128*n - 96)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(8*n^3 - 32*n^2 + 39*n - 15 + (16*n^2 - 50*n + 34)*x + (8*n - 8)*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] = ((56*n^3 - 208*n^2 + 197*n - 17 + (96*n^3 - 272*n^2 + 30*n + 242)*x + (192*n^2 - 576*n + 432)*x^2 + (96*n - 160)*x^3)*v[i - 1] - (8*n^2 - 12*n + 1 + (48*n^2 - 56*n - 12)*x + (64*n^2 - 92)*x^2 + (128*n - 96)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(8*n^3 - 32*n^2 + 39*n - 15 + (16*n^2 - 50*n + 34)*x + (8*n - 8)*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

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