Real-rooted atlas

A091044 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n^{2} + n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(-4\,x - 2\,x^{2} + 4\,n + 4\,n x - 4\,n x^{2} + 8\,n^{2} + 28\,n^{2} x + 18\,n^{2} x^{2} + 8\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} - 4\,x^{3} - 4\,n x - 16\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2166436116
3643844801281056
42562048403225604009296
5102410240276482867211200115279936
6409649152168960245760161280430083136675392
7163842293769584641802240168960077414415052881925628928

All 15 rows

Technical details
Node id
n-690542817ffd3840 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4f2817c663d253d1

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:      4,       8;
  2:     16,      64,       36;
  3:     64,     384,      480,      128;
  4:    256,    2048,     4032,     2560,       400;
  5:   1024,   10240,    27648,    28672,     11200,     1152;
  6:   4096,   49152,   168960,   245760,    161280,    43008,     3136;
  7:  16384,  229376,   958464,  1802240,   1689600,   774144,   150528,     8192;
  8:  65536, 1048576,  5160960, 11927552,  14643200,  9732096,  3311616,   491520,   20736;
  9: 262144, 4718592, 26738688, 73400320, 111820800, 98402304, 49201152, 12976128, 1520640, 51200;
%F Formula
The row polynomials satisfy (2*n^2+n+(2*n^3+n^2)*x)*P(n,x) = (8*n^2+4*n+(8*n^3+28*n^2+4*n-4)*x+(12*n^3+18*n^2-4*n-2)*x^2)*P(n-1,x) - ((4*n+2)*x+(4*n^2+16*n+6)*x^2+(8*n^2+16*n+4)*x^3)*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[((8*n^2 + 4*n + (8*n^3 + 28*n^2 + 4*n - 4)*x + (12*n^3 + 18*n^2 - 4*n - 2)*x^2)*P[n - 1, x] - ((4*n + 2)*x + (4*n^2 + 16*n + 6)*x^2 + (8*n^2 + 16*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^2 + n + (2*n^3 + n^2)*x)]];
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] = ((8*n^2 + 4*n + (8*n^3 + 28*n^2 + 4*n - 4)*x + (12*n^3 + 18*n^2 - 4*n - 2)*x^2)*v[i - 1] - ((4*n + 2)*x + (4*n^2 + 16*n + 6)*x^2 + (8*n^2 + 16*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^2 + n + (2*n^3 + n^2)*x)); 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