Real-rooted atlas

A008292 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,n x^{2} + n^{2} + 4\,n^{2} x + 3\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-n x^{2} - n x^{3} + n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x - n - 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x + x^{2} - n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3612321
42472424142
512048045012051175
6720360046802160310611476
7504030240504003360090307567129073
84032028224057456050400020412034776177881641802

All 15 rows

Technical details
Node id
n-94b24c22fc28335f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6080e9ebf914aab7

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       2,        2;
   3:       6,       12,        3;
   4:      24,       72,       42,         4;
   5:     120,      480,      450,       120,        5;
   6:     720,     3600,     4680,      2160,      310,        6;
   7:    5040,    30240,    50400,     33600,     9030,      756,       7;
   8:   40320,   282240,   574560,    504000,   204120,    34776,    1778,      8;
   9:  362880,  2903040,  6985440,   7620480,  4170600,  1118880,  127050,   4080,    9;
  10: 3628800, 32659200, 90720000, 118540800, 82177200, 30618000, 5729640, 447840, 9198, 10;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3-n^2+(2*n^3-4*n^2)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (n^2*x+(2*n^2-n)*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2-n+(2*n^2-4*n+2)*x)*P(n-1,x) - ((n-1)*x+(n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (n^2*x + (2*n^2 - n)*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - n + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] - ((n - 1)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(n^3 - n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (n^2*x + (2*n^2 - n)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - n + (2*n^2 - 4*n + 2)*x)*v[i - 1] - ((n - 1)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.