Real-rooted atlas

A086645 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-8 - 4\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(8\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31632856
46428819232576
52562048256010241286016
61024128002560017920512051262976
7409673728215040229376110592245762048659456
81638440140816056322408448172032063078411468881926905856
96553620971521101004822020096216268801153433634078725242883276872318976
10262144106168327077888018166579223357030416868966471565312176947202359296131072757334016
11104857652428800432537600138412032022491955202099249152119275520041943040089128960104857605242887930904576
124194304253755392253755392098964602881979292057623091740672167939932167843348480235300454443830476846137344209715283053510656
131677721612079595521439485132867175972864161942077440230317621248207634825216123211874304487713669121275068416021139292162013265928388608869747654656
146710886456706990087938978611243664382361612475537817602120841428992231364519526416907407196168453703598082900780646406717597286410032775168872415232335544329108115685376
152684354562630667468842748346368027358941675529135932309504182718646190082367037007462420809116549120127455838863365498095009792166389717401634574486732846976204800375809638413421772895381425750016
Technical details
Node id
n-584607888f6a6ba8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4a6fdd8c1c3f626a

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,        2;
   3:     16,       32,        8;
   4:     64,      288,      192,        32;
   5:    256,     2048,     2560,      1024,       128;
   6:   1024,    12800,    25600,     17920,      5120,       512;
   7:   4096,    73728,   215040,    229376,    110592,     24576,     2048;
   8:  16384,   401408,  1605632,   2408448,   1720320,    630784,   114688,     8192;
   9:  65536,  2097152, 11010048,  22020096,  21626880,  11534336,  3407872,   524288,   32768;
  10: 262144, 10616832, 70778880, 181665792, 233570304, 168689664, 71565312, 17694720, 2359296, 131072;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(2*n-4)*x)*P(n-1,x) + (8*x+2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (2*n - 4)*x)*P[n - 1, x] + (8*x + 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 8 + (2*n - 4)*x)*v[i - 1] + (8*x + 2*x^2)*deriv(v[i - 1], x))/(n - 2)); 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