Real-rooted atlas

A207606 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(288 + 384\,n + 128\,n^{2}\right)\,P_{n}(x) = \left(1152 + 1104\,x + 228\,x^{2} + 1536\,n + 1600\,n x + 380\,n x^{2} + 15\,n x^{3} + 512\,n^{2} + 576\,n^{2} x + 152\,n^{2} x^{2} + 15\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-384\,x - 720\,x^{2} - 228\,x^{3} - 256\,n x - 480\,n x^{2} - 224\,n x^{3} - 30\,n x^{4}\right)\,P_{n-1}'(x) + \left(48\,x^{3} + 72\,x^{4} + 15\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 32 + 44\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
032441389
112827217634610
251214721488620894181
320487424102406672204023328657
48192358406272055936266886432610196418
532768167936356352404480264320991201969615971346269
613107277004819210242651136220800011335043494565904441819227465
752428834734089961472161996801642905610753536453401611854241741601094663245986

All 15 rows

Technical details
Node id
n-cb8da48eb729d7a8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d4f5100b4a0a6575

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:      32,       44,        13;
  1:     128,      272,       176,        34;
  2:     512,     1472,      1488,       620,        89;
  3:    2048,     7424,     10240,      6672,      2040,       233;
  4:    8192,    35840,     62720,     55936,     26688,      6432,       610;
  5:   32768,   167936,    356352,    404480,    264320,     99120,     19696,      1597;
  6:  131072,   770048,   1921024,   2651136,   2208000,   1133504,    349456,     59044,     4181;
  7:  524288,  3473408,   9961472,  16199680,  16429056,  10753536,   4534016,   1185424,   174160,    10946;
  8: 2097152, 15466496,  50135040,  93913088, 112394240,  89650176,  48188672,  17210240,  3903456,   507220,   28657;
  9: 8388608, 68157440, 246415360, 522584064, 721354752, 679583744, 445276160, 202653440, 62710528, 12553824, 1462128, 75025;
%F Formula
The row polynomials satisfy (128*n^2+384*n+288)*P(n,x) = (512*n^2+1536*n+1152+(576*n^2+1600*n+1104)*x+(152*n^2+380*n+228)*x^2+(15*n^2+15*n)*x^3)*P(n-1,x) - ((256*n+384)*x+(480*n+720)*x^2+(224*n+228)*x^3+30*n*x^4)*d/dx P(n-1,x) + (48*x^3+72*x^4+15*x^5)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 32 + 44*x + 13*x^2.
%t Mathematica
P[0, x_] := 32 + 44*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((512*n^2 + 1536*n + 1152 + (576*n^2 + 1600*n + 1104)*x + (152*n^2 + 380*n + 228)*x^2 + (15*n^2 + 15*n)*x^3)*P[n - 1, x] - ((256*n + 384)*x + (480*n + 720)*x^2 + (224*n + 228)*x^3 + 30*n*x^4)*D[P[n - 1, x], x] + (48*x^3 + 72*x^4 + 15*x^5)*D[P[n - 1, x], {x, 2}])/(128*n^2 + 384*n + 288)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 32 + 44*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((512*n^2 + 1536*n + 1152 + (576*n^2 + 1600*n + 1104)*x + (152*n^2 + 380*n + 228)*x^2 + (15*n^2 + 15*n)*x^3)*v[i - 1] - ((256*n + 384)*x + (480*n + 720)*x^2 + (224*n + 228)*x^3 + 30*n*x^4)*deriv(v[i - 1], x) + (48*x^3 + 72*x^4 + 15*x^5)*deriv(deriv(v[i - 1], x), x))/(128*n^2 + 384*n + 288)); 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