Real-rooted atlas

A103327 → basis A013611 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-24 - 20\,x + 32\,n + 24\,n x\right)\,P_{n-1}(x) + \left(-20 - 36\,x - 16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
2808016176
38321280576642752
487041817613056358425643776
591136240640238592107520204801024699392
6954368304742438584322433024778240110592409611186176
7999424037437440577044484685824021217280516096057344016384178962432
8104660992449904640817430528812777472477429760166068224321126402883584655362863333376
91096024064531732889611127226368130975006729469165568430099660812000952321903165441415577626214445813071872
101147771289662028513280146914934784199827128320171683348480964260003843534225408081537269761085276160681574401048576733008101376
111201961697287160138301441893480792064292253650124829101549158401949189537792887174987776270318698496527349841925997854720322961408419430411728125427712
121258710630400819399229440023930685358080413273161728004681011036160036437596569600198415745024007536996515840195160965120032757514240032296140800150994944016777216187649990066176
131318138884915293095093862400297588888698880568565086289920721960514355200640659467796480406468322918400185274822819840600265300377601343519457280019672963481601701209702406979321856671088643002399773949952
14138037296103424105120962025881636505893260492807645369278136320107577200135372801072143559294976077704655693414404136520583741440161264210804736045314991980544088864215531520114809844531208793945538563194381926426843545648038396114763776

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-16 - 32\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(12 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 4\,x\)
Technical details
Node id
n-208323846157519b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-9cb3ec373d16ebdc)
Fingerprint
f6-5870c1c52d074a69

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:          8,          4;
  2:         80,         80,          16;
  3:        832,       1280,         576,          64;
  4:       8704,      18176,       13056,        3584,        256;
  5:      91136,     240640,      238592,      107520,      20480,       1024;
  6:     954368,    3047424,     3858432,     2433024,     778240,     110592,       4096;
  7:    9994240,   37437440,    57704448,    46858240,   21217280,    5160960,     573440,     16384;
  8:  104660992,  449904640,   817430528,   812777472,  477429760,  166068224,   32112640,   2883584,    65536;
  9: 1096024064, 5317328896, 11127226368, 13097500672, 9469165568, 4300996608, 1200095232, 190316544, 14155776, 262144;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (32*n-24+(24*n-20)*x)*P(n-1,x) - (20+36*x+16*x^2)*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 - 24 + (24*n - 20)*x)*P[n - 1, x] - (20 + 36*x + 16*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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 - 24 + (24*n - 20)*x)*v[i - 1] - (20 + 36*x + 16*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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