Real-rooted atlas

A133214 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(16 + 16\,x + 32\,n + 48\,n x + 14\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-3 - 18\,x - 24\,x^{2} - 2\,n - 12\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 32 + 32\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0323264
1228480192900
2153649924608102412160
3100204416065280358405120160420
464032356640729600660480245760245762081088
54033962716896711648090675205698560154828811468826665828
62514944198732806352281610479616094044160440401929175040524288338490880
715554628141083136532675584108096307212683059208587837443137863685190451223592964265416260

All 15 rows

Technical details
Node id
n-bf8088359dccab94 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b91fbc6aa50ad79c

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        32,         32;
  1:       228,        480,         192;
  2:      1536,       4992,        4608,        1024;
  3:     10020,      44160,       65280,       35840,         5120;
  4:     64032,     356640,      729600,      660480,       245760,        24576;
  5:    403396,    2716896,     7116480,     9067520,      5698560,      1548288,       114688;
  6:   2514944,   19873280,    63522816,   104796160,     94044160,     44040192,      9175040,      524288;
  7:  15554628,  141083136,   532675584,  1080963072,   1268305920,    858783744,    313786368,    51904512,     2359296;
  8:  95600160,  978850080,  4264151040, 10282782720,  14946140160,  13346242560,   7129006080,  2099773440,   283115520,   10485760;
  9: 584585892, 6669162720, 32932152000, 92056842240, 159861166080, 177964351488, 126333419520, 54932275200, 13365411840, 1499463680, 46137344;
%F Formula
The row polynomials satisfy (2*n^2+4*n+2)*P(n,x) = (14*n^2+32*n+16+(24*n^2+48*n+16)*x)*P(n-1,x) - (2*n+3+(12*n+18)*x+(16*n+24)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 32 + 32*x.
%t Mathematica
P[0, x_] := 32 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^2 + 32*n + 16 + (24*n^2 + 48*n + 16)*x)*P[n - 1, x] - (2*n + 3 + (12*n + 18)*x + (16*n + 24)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 4*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 32 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^2 + 32*n + 16 + (24*n^2 + 48*n + 16)*x)*v[i - 1] - (2*n + 3 + (12*n + 18)*x + (16*n + 24)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 4*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