Real-rooted atlas

A114656 → basis A007318 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(32\,x^{2} - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-4 - 4\,x + 8\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2448
320401676
411233628864800
56722688358417922568992
64224211203840030720102401024105728
7274561647363801604224002304005529640961284544
8183040128128035875845125120394240015769602867201638415999488

All 15 rows

Technical details
Node id
n-9627f9ea052c9c8b · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-665cb75b242a18c2

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,        4;
   3:      20,       40,        16;
   4:     112,      336,       288,        64;
   5:     672,     2688,      3584,      1792,       256;
   6:    4224,    21120,     38400,     30720,     10240,      1024;
   7:   27456,   164736,    380160,    422400,    230400,     55296,      4096;
   8:  183040,  1281280,   3587584,   5125120,   3942400,   1576960,    286720,    16384;
   9: 1244672,  9957376,  32800768,  57401344,  57401344,  32800768,  10092544,  1441792,   65536;
  10: 8599552, 77395968, 293289984, 608305152, 751435776, 563576832, 250478592, 61341696, 7077888, 262144;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(16*n-32)*x^2*P(n-2,x) + (8*n-4+(8*n-4)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n - 32)*x^2*P[n - 2, x] + (8*n - 4 + (8*n - 4)*x)*P[n - 1, 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; v[2] = 4 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(16*n - 32)*x^2*v[i - 2] + (8*n - 4 + (8*n - 4)*x)*v[i - 1])/(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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023