Real-rooted atlas

A059364 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2614626
324928724227
41206529875841202463
572051121081210084437072031818
6504044424121692157084106220365045040476004
7403204259521439988239246422242451184280338436403208086005

All 15 rows

Technical details
Node id
n-31270249172ef2fe · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d28163317426c881

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,        2;
  2:       6,       14,         6;
  3:      24,       92,        87,        24;
  4:     120,      652,       987,       584,       120;
  5:     720,     5112,     10812,     10084,      4370,       720;
  6:    5040,    44424,    121692,    157084,    106220,     36504,      5040;
  7:   40320,   425952,   1439988,   2392464,   2224245,   1184280,    338436,     40320;
  8:  362880,  4478688,  18071316,  36892048,  43942845,  32011272,  14099204,   3457152,   362880;
  9: 3628800, 51318720, 241175448, 585776272, 854463990, 794695446, 477110396, 179591904, 38632464, 3628800;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(2*n^3-2*n^2)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n^2-1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(n - 1)); 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. [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.