Real-rooted atlas

A123519 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(3 + 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(21 + 24\,x + 19\,n + 20\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(6 + 16\,x + 8\,x^{2} + 3\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 28 + 32\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0283260
1152352192696
27442592288010247240
33436159682688019456512070860
4152848876820044822016011776024576666996
5662244613761307520193024015667206635521146886110320
628142422865287797888144742401580032010149888355532852428854869904
7117819610936128436032009755545613400576011577753661472768183500802359296485238420

All 15 rows

Technical details
Node id
n-63684c7cc5efa4d2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-cda21f326eecd7ab

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:       28,        32;
  1:      152,       352,        192;
  2:      744,      2592,       2880,       1024;
  3:     3436,     15968,      26880,      19456,       5120;
  4:    15284,     88768,     200448,     220160,     117760,      24576;
  5:    66224,    461376,    1307520,    1930240,    1566720,     663552,     114688;
  6:   281424,   2286528,    7797888,   14474240,   15800320,   10149888,    3555328,     524288;
  7:  1178196,  10936128,   43603200,   97555456,  134005760,  115777536,   61472768,   18350080,    2359296;
  8:  4874444,  50883552,  232231680,  608297984, 1008000000, 1096187904,  782630912,  353894400,   92012544,  10485760;
  9: 19973192, 231593632, 1190624832, 3574205440, 6939484160, 9107374080, 8184594432, 4974968832, 1958215680, 450887680, 46137344;
%F Formula
The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (4*n^2+19*n+21+(4*n^2+20*n+24)*x)*P(n-1,x) + (3*n+6+(8*n+16)*x+(4*n+8)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 28 + 32*x.
%t Mathematica
P[0, x_] := 28 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 19*n + 21 + (4*n^2 + 20*n + 24)*x)*P[n - 1, x] + (3*n + 6 + (8*n + 16)*x + (4*n + 8)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 28 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 19*n + 21 + (4*n^2 + 20*n + 24)*x)*v[i - 1] + (3*n + 6 + (8*n + 16)*x + (4*n + 8)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 5*n + 3)); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023