Real-rooted atlas

A207823 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(8\,n\right)\,P_{n}(x) = \left(-20\,x - 21\,x^{2} + 32\,n + 60\,n x + 21\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 40\,x^{2} - 21\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1459
216402480
364240292115711
42561280235218805516319
510246400157441904011312264056160
6409630720947201536001380966524012649499121
71638414336053145610816001304640932400365484606054435929
8655366553602838528695296010531840100992005985840200460029037639424240
92621442949120146145284186112076360704919744007313472037015200108190201391275350382231
10104857613107200731381762398617605119098887427481607418790405036352002223662405765560066659993114015839
1141943045767168035782656013221888003232333824548868096066048962565631961600333450624013054384003041229763193872027675760320
12167772162516582401718616064706478080019467141120378765312005335107993654809579520407559539202139064960075213176641590714160153027601245967827041
136710886410905190408128561152367945318401128346419202475196416003995546419204805050368004304668170242837104064001337040124164266400656082615582287331992852186034683049
1426843545646976204803795006259218756927488063359261081615471869952002816349634560388191682560040710413721603232366131200191261212774481778543872023884494804842649874120351296882419428344320400
Technical details
Node id
n-beaf9b3575bb564b · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4a3147452b8a179e

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:      4,       5;
  2:     16,      40,       24;
  3:     64,     240,      292,      115;
  4:    256,    1280,     2352,     1880,      551;
  5:   1024,    6400,    15744,    19040,    11312,     2640;
  6:   4096,   30720,    94720,   153600,   138096,    65240,    12649;
  7:  16384,  143360,   531456,  1081600,  1304640,   932400,   365484,    60605;
  8:  65536,  655360,  2838528,  6952960, 10531840, 10099200,  5985840,  2004600,   290376;
  9: 262144, 2949120, 14614528, 41861120, 76360704, 91974400, 73134720, 37015200, 10819020, 1391275;
%F Formula
The row polynomials satisfy 8*n*P(n,x) = (32*n+(60*n-20)*x+(21*n-21)*x^2)*P(n-1,x) - (16*x+40*x^2+21*x^3)*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[((32*n + (60*n - 20)*x + (21*n - 21)*x^2)*P[n - 1, x] - (16*x + 40*x^2 + 21*x^3)*D[P[n - 1, x], x])/(8*n)]];
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] = ((32*n + (60*n - 20)*x + (21*n - 21)*x^2)*v[i - 1] - (16*x + 40*x^2 + 21*x^3)*deriv(v[i - 1], x))/(8*n)); 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

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