Real-rooted atlas

A069466 → basis A007318 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-12 - 48\,x - 48\,x^{2} + 32\,n + 128\,n x + 128\,n x^{2} - 16\,n^{2} - 64\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(6 + 4\,x - 24\,n - 16\,n x + 24\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2789624198
31260228012001604900
4224705376043680134401120134470
54241161262520139104068544014112080643912300
683058362958278441191920280896009535680141926459136118184220
716692904869222753611655483841017656640486245760122234112138378244392963665118600

All 15 rows

Technical details
Node id
n-6ebbdbbb1ecce33b · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-e7d8403424fc1d9b

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:           6,            4;
  2:          78,           96,           24;
  3:        1260,         2280,         1200,           160;
  4:       22470,        53760,        43680,         13440,         1120;
  5:      424116,      1262520,      1391040,        685440,       141120,         8064;
  6:     8305836,     29582784,     41191920,      28089600,      9535680,      1419264,        59136;
  7:   166929048,    692227536,   1165548384,    1017656640,    486245760,    122234112,     13837824,      439296;
  8:  3419932230,  16183664640,  31971139200,   34064110080,  21073852800,   7564677120,   1476034560,   131788800,    3294720;
  9: 71109813060, 378133006680, 857432759040, 1079111953920, 821194133760, 385406461440, 108734545920, 17027112960, 1232225280, 24893440;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -(16*n^2-32*n+12+(64*n^2-128*n+48)*x+(64*n^2-128*n+48)*x^2)*P(n-2,x) + (24*n^2-24*n+6+(16*n^2-16*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n^2 - 32*n + 12 + (64*n^2 - 128*n + 48)*x + (64*n^2 - 128*n + 48)*x^2)*P[n - 2, x] + (24*n^2 - 24*n + 6 + (16*n^2 - 16*n + 4)*x)*P[n - 1, x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(16*n^2 - 32*n + 12 + (64*n^2 - 128*n + 48)*x + (64*n^2 - 128*n + 48)*x^2)*v[i - 2] + (24*n^2 - 24*n + 6 + (16*n^2 - 16*n + 4)*x)*v[i - 1])/(n^2)); 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. [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