Real-rooted atlas

A053117 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−3, −1)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(2\,n + n x\right)\,P_{n-1}(x) + \left(3 + 4\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31516435
45692488204
5209464372128161189
6780218223681248320326930
72911982413464960037607686440391
81086442936711366408033920105601792128235416
9405451836483569523889922600801093122822440962561372105
101513167727461723968220579217843849468483297287270492165127997214
115647193209968808441211882112112890407246848318438494617618201620480102446611179
122107560131965643703361661491480671360005061862426901504100807682611200445440450562048271669860
137865521537914081664431803081510403802379203294996482056588809317990430424320698368010700809830440961583407981
142935452421770011073632192015039990402070117760202785580814547333127739566083054458888833792018202624253132821299281929228778026
151095525758757180803214244720718048896010908231200119221591049668985984591211315227392724489572597762484162564643225659105284587521638453789260175

Other recurrences

Image under the basis map A013609 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)
Technical details
Node id
n-5c55f31aa1da8314 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A013609 (equal by e-e29e2a5899c4b62d)
Fingerprint
f6-c761722d499074dd

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:     15,     16,       4;
   4:     56,     92,      48,       8;
   5:    209,    464,     372,     128,      16;
   6:    780,   2182,    2368,    1248,     320,     32;
   7:   2911,   9824,   13464,    9600,    3760,    768,     64;
   8:  10864,  42936,   71136,   64080,   33920,  10560,   1792,   128;
   9:  40545, 183648,  356952,  388992,  260080, 109312,  28224,  4096,  256;
  10: 151316, 772746, 1723968, 2205792, 1784384, 946848, 329728, 72704, 9216, 512;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n+n*x)*P(n-1,x) + (3+4*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + n*x)*P[n - 1, x] + (3 + 4*x + x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n + n*x)*v[i - 1] + (3 + 4*x + x^2)*deriv(v[i - 1], x))/(n - 1)); 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