Real-rooted atlas

A116412 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(64 + 56\,x + 16\,x^{2} - 114\,n - 60\,n x + 56\,n^{2} - 8\,n^{2} x - 16\,n^{2} x^{2} - 6\,n^{3} + 12\,n^{3} x\right)\,P_{n}(x) = \left(104 + 160\,x + 32\,x^{2} - 420\,n - 888\,n x - 528\,n x^{2} - 96\,n x^{3} + 304\,n^{2} + 512\,n^{2} x + 160\,n^{2} x^{2} - 48\,n^{3} - 48\,n^{3} x\right)\,P_{n-1}(x) + \left(26 + 92\,x + 88\,x^{2} + 16\,x^{3} - 91\,n - 374\,n x - 492\,n x^{2} - 232\,n x^{3} - 32\,n x^{4} + 39\,n^{2} + 138\,n^{2} x + 132\,n^{2} x^{2} + 24\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
216824
3849624204
4380696384641524
5159440403552128016010626
66384207122524814400384038470968
724782980881535521212805088010752896460230
8940604396408423048550404956801639682867220482921412
93509881892376429364853420163978720181440049459273728460818245076
1012924327896960207192963057696027924480163457286128640141926418432010240112498320
11470853432156936958195441637740801775496001274192646125414419470336391680045056022528686542326
1217004192128371632428339616832669440104645088089174131252534272021384806458936320104755201081344491524154310192
136095983450412273618624240964059540160580801104057402443524015323648200180736070569216017155072027303936255590410649624959642442
1421718602419525959447913095008191199931523070100800034563182976280459468801659793612871612375042224302080483450880696483845963776229376149055776112
Technical details
Node id
n-8715736404ed5023 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8925b7cc248dd831

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      2;
  2:     16,       8;
  3:     84,      96,      24;
  4:    380,     696,     384,      64;
  5:   1594,    4040,    3552,    1280,     160;
  6:   6384,   20712,   25248,   14400,    3840,     384;
  7:  24782,   98088,  153552,  121280,   50880,   10752,    896;
  8:  94060,  439640,  842304,  855040,  495680,  163968,  28672,  2048;
  9: 350988, 1892376, 4293648, 5342016, 3978720, 1814400, 494592, 73728, 4608;
%F Formula
The row polynomials satisfy -(6*n^3-56*n^2+114*n-64-(12*n^3-8*n^2-60*n+56)*x+(16*n^2-16)*x^2)*P(n,x) = -(48*n^3-304*n^2+420*n-104+(48*n^3-512*n^2+888*n-160)*x-(160*n^2-528*n+32)*x^2+96*n*x^3)*P(n-1,x) + (39*n^2-91*n+26+(138*n^2-374*n+92)*x+(132*n^2-492*n+88)*x^2+(24*n^2-232*n+16)*x^3-32*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(48*n^3 - 304*n^2 + 420*n - 104 + (48*n^3 - 512*n^2 + 888*n - 160)*x - (160*n^2 - 528*n + 32)*x^2 + 96*n*x^3)*P[n - 1, x] + (39*n^2 - 91*n + 26 + (138*n^2 - 374*n + 92)*x + (132*n^2 - 492*n + 88)*x^2 + (24*n^2 - 232*n + 16)*x^3 - 32*n*x^4)*D[P[n - 1, x], x])/(-6*n^3 + 56*n^2 - 114*n + 64 + (12*n^3 - 8*n^2 - 60*n + 56)*x - (16*n^2 - 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n^3 - 304*n^2 + 420*n - 104 + (48*n^3 - 512*n^2 + 888*n - 160)*x - (160*n^2 - 528*n + 32)*x^2 + 96*n*x^3)*v[i - 1] + (39*n^2 - 91*n + 26 + (138*n^2 - 374*n + 92)*x + (132*n^2 - 492*n + 88)*x^2 + (24*n^2 - 232*n + 16)*x^3 - 32*n*x^4)*deriv(v[i - 1], x))/(-6*n^3 + 56*n^2 - 114*n + 64 + (12*n^3 - 8*n^2 - 60*n + 56)*x - (16*n^2 - 16)*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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.