Real-rooted atlas

A130534 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2} x\right)\,P_{n}(x) = \left(-2\,x + x^{2} - n - 2\,n x + 2\,n x^{2} - n^{2} + n^{2} x + n^{2} x^{2} + n^{3} x\right)\,P_{n-1}(x) + \left(-x + x^{2} - n x + 2\,n x^{2} + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
2624939
32414410816292
41209601080320252505
572072001080048007503624306
65040604801134006720015750151249263431
7403205644801270080940800294000423362744643154824

All 15 rows

Technical details
Node id
n-3202cf200924ca0a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-eb1d45955591e332

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:       2,        4;
  2:       6,       24,         9;
  3:      24,      144,       108,        16;
  4:     120,      960,      1080,       320,       25;
  5:     720,     7200,     10800,      4800,      750,       36;
  6:    5040,    60480,    113400,     67200,    15750,     1512,      49;
  7:   40320,   564480,   1270080,    940800,   294000,    42336,    2744,     64;
  8:  362880,  5806080,  15240960,  13547520,  5292000,  1016064,   98784,   4608,   81;
  9: 3628800, 65318400, 195955200, 203212800, 95256000, 22861440, 2963520, 207360, 7290, 100;
%F Formula
The row polynomials satisfy -(n-n^2*x)*P(n,x) = -(n^2+n-(n^3+n^2-2*n-2)*x-(n^2+2*n+1)*x^2)*P(n-1,x) - ((n+1)*x-(n^2+2*n+1)*x^2)*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[(-(n^2 + n - (n^3 + n^2 - 2*n - 2)*x - (n^2 + 2*n + 1)*x^2)*P[n - 1, x] - ((n + 1)*x - (n^2 + 2*n + 1)*x^2)*D[P[n - 1, x], x])/(-n + 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] = (-(n^2 + n - (n^3 + n^2 - 2*n - 2)*x - (n^2 + 2*n + 1)*x^2)*v[i - 1] - ((n + 1)*x - (n^2 + 2*n + 1)*x^2)*deriv(v[i - 1], x))/(-n + 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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.