Real-rooted atlas

A078812 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-8 + 8\,n\right)\,P_{n}(x) = \left(-32 - 48\,x - 10\,x^{2} + 32\,n + 36\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 24\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
31624848
46414410021329
5256768816360552255
61024384055043744120014415456
74096184323328030720152163816377105937
816384860161873922188801475205716811756987726103
9655363932161003520141926412134406405122033123540025844976784
10262144176947251773448601600892416059834882587264694368104780676534111385
111048576786432025952256495452166049792049287168270860809898752229891230600017711233802911
12419430434603008127139840274268160385449984370040832247281664114869760363046407426368884064463681602508992
13167772161509949446113198081470627840233865216025869680642038751232115203686446274176012872064023516448253166412139310983760033
146710886465431142428940697607681867776136393523201708956057615530459136103543603205059795968178823808044392307273249632719650031781175283811239
1526843545628185722881352243609639258685440769916928001078056714241110518661128542185062449263513600211803033606677664768149604710422501844820329080832040516002918640
Technical details
Node id
n-41fdaa260044e0da · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-882d99bdfc839db5

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:      4,       3;
   3:     16,      24,       8;
   4:     64,     144,     100,      21;
   5:    256,     768,     816,     360,      55;
   6:   1024,    3840,    5504,    3744,    1200,     144;
   7:   4096,   18432,   33280,   30720,   15216,    3816,     377;
   8:  16384,   86016,  187392,  218880,  147520,   57168,   11756,    987;
   9:  65536,  393216, 1003520, 1419264, 1213440,  640512,  203312,  35400,   2584;
  10: 262144, 1769472, 5177344, 8601600, 8924160, 5983488, 2587264, 694368, 104780, 6765;
%F Formula
The row polynomials satisfy (8*n-8)*P(n,x) = (32*n-32+(36*n-48)*x+(5*n-10)*x^2)*P(n-1,x) - (16*x+24*x^2+5*x^3)*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[((32*n - 32 + (36*n - 48)*x + (5*n - 10)*x^2)*P[n - 1, x] - (16*x + 24*x^2 + 5*x^3)*D[P[n - 1, x], x])/(8*n - 8)]];
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] = ((32*n - 32 + (36*n - 48)*x + (5*n - 10)*x^2)*v[i - 1] - (16*x + 24*x^2 + 5*x^3)*deriv(v[i - 1], x))/(8*n - 8)); 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. [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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023