Real-rooted atlas

A207824 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-16\,n - 3\,n x + n^{3} x^{2}\right)\,P_{n}(x) = \left(-48 - 33\,x - 6\,x^{2} - 48\,n - 41\,n x - 6\,n x^{2} + n x^{3} - 8\,n^{2} x + 3\,n^{2} x^{2} + 2\,n^{2} x^{3} + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-64 - 72\,x - 26\,x^{2} - 3\,x^{3} + 8\,n x^{2} + 6\,n x^{3} + n x^{4} + 8\,n^{2} x^{2} + 6\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
235481295
320442421632876
4118933122556768807905
56930242042505611392240019270174
640391169632220392134400428006912448614975
72354161155152180619213827205904001440001881610245333720
81372105770294414082216129784326956400227404844844849152230445866049
97997214505509321057959361140404487359408030195648797798413189121244165120391600690
1046611179327591504772346196953386752719709200355525632117875968260997123711744307200112643323176351
1127166986021014428085511055752766586992066281280003830260224152883763242471014480870400100864007434242457628053699140
1215834079811336774478438598316740597379814405821373360038533542912179924236806016647168143710848023992320026639360176947253248235749292065
13922877802684438657820266182079040453690863360491999227200367083957888196337304576768007127042207923660846230272006869237766871449641533441146881973223750726
14537892601755301793142401811810130960337222607616040280087880003344768349696201656063884890355861094430383991475276563210240142655631361909260288173748224963379224576016457662745215
Technical details
Node id
n-dcd3f892e955a974 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a7caceb584b0a586

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:      35,       48,        12;
  3:     204,      424,       216,        32;
  4:    1189,     3312,      2556,       768,       80;
  5:    6930,    24204,     25056,     11392,     2400,      192;
  6:   40391,   169632,    220392,    134400,    42800,     6912,     448;
  7:  235416,  1155152,   1806192,   1382720,   590400,   144000,   18816,    1024;
  8: 1372105,  7702944,  14082216,  12978432,  6956400,  2274048,  448448,   49152,   2304;
  9: 7997214, 50550932, 105795936, 114040448, 73594080, 30195648, 7977984, 1318912, 124416, 5120;
%F Formula
The row polynomials satisfy -(16*n+3*n*x-n^3*x^2)*P(n,x) = -(48*n+48+(8*n^2+41*n+33)*x-(3*n^3+3*n^2-6*n-6)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (64+72*x-(8*n^2+8*n-26)*x^2-(6*n^2+6*n-3)*x^3-(n^2+n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(48*n + 48 + (8*n^2 + 41*n + 33)*x - (3*n^3 + 3*n^2 - 6*n - 6)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (64 + 72*x - (8*n^2 + 8*n - 26)*x^2 - (6*n^2 + 6*n - 3)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-16*n - 3*n*x + n^3*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(48*n + 48 + (8*n^2 + 41*n + 33)*x - (3*n^3 + 3*n^2 - 6*n - 6)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (64 + 72*x - (8*n^2 + 8*n - 26)*x^2 - (6*n^2 + 6*n - 3)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-16*n - 3*n*x + n^3*x^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

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