Real-rooted atlas

A178820 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - n x + 3\,n^{2} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(6 + 18\,x + 18\,x^{2} + 6\,x^{3} + 2\,n + 24\,n x + 36\,n x^{2} + 14\,n x^{3} + 6\,n^{2} x + 16\,n^{2} x^{2} + 10\,n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
183240
240320360720
31601920432025608960
4560896030240358401400089600
517923584016128028672022400064512774144
653761290247257601720320201600011612162634246021120
715360430080290304086016001344000011612160526848098304043253760

All 15 rows

Technical details
Node id
n-5eb24a493cbaafd0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:      8,      32;
  2:     40,     320,      360;
  3:    160,    1920,     4320,      2560;
  4:    560,    8960,    30240,     35840,     14000;
  5:   1792,   35840,   161280,    286720,    224000,     64512;
  6:   5376,  129024,   725760,   1720320,   2016000,   1161216,    263424;
  7:  15360,  430080,  2903040,   8601600,  13440000,  11612160,   5268480,    983040;
  8:  42240, 1351680, 10644480,  37847040,  73920000,  85155840,  57953280,  21626880,  3421440;
  9: 112640, 4055040, 36495360, 151388160, 354816000, 510935040, 463626240, 259522560, 82114560, 11264000;
%F Formula
The row polynomials satisfy (n+(3*n^2-n)*x+n^3*x^2)*P(n,x) = (2*n+6+(6*n^2+24*n+18)*x+(2*n^3+16*n^2+36*n+18)*x^2+(2*n^3+10*n^2+14*n+6)*x^3)*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[((2*n + 6 + (6*n^2 + 24*n + 18)*x + (2*n^3 + 16*n^2 + 36*n + 18)*x^2 + (2*n^3 + 10*n^2 + 14*n + 6)*x^3)*P[n - 1, x])/(n + (3*n^2 - 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] = ((2*n + 6 + (6*n^2 + 24*n + 18)*x + (2*n^3 + 16*n^2 + 36*n + 18)*x^2 + (2*n^3 + 10*n^2 + 14*n + 6)*x^3)*v[i - 1])/(n + (3*n^2 - 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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.