Real-rooted atlas

A156992 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 3\,x + 3\,x^{2} + x^{3} + n + 6\,n x + 8\,n x^{2} + 3\,n x^{3} + 3\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
264854108
3242886483841344
4120192064807680300019200
572014400648001152009000025920311040
650401209606804001612800189000010886402469605644800
74032011289607620480225792003528000030481920138297602580480113541120

All 15 rows

Technical details
Node id
n-8e5364d2ad0e7f06 · 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:       2,         8;
  2:       6,        48,         54;
  3:      24,       288,        648,        384;
  4:     120,      1920,       6480,       7680,        3000;
  5:     720,     14400,      64800,     115200,       90000,       25920;
  6:    5040,    120960,     680400,    1612800,     1890000,     1088640,      246960;
  7:   40320,   1128960,    7620480,   22579200,    35280000,    30481920,    13829760,    2580480;
  8:  362880,  11612160,   91445760,  325140480,   635040000,   731566080,   497871360,  185794560,   29393280;
  9: 3628800, 130636800, 1175731200, 4877107200, 11430720000, 16460236800, 14936140800, 8360755200, 2645395200, 362880000;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (n+1+(3*n^2+6*n+3)*x+(n^3+6*n^2+8*n+3)*x^2+(n^3+3*n^2+3*n+1)*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[((n + 1 + (3*n^2 + 6*n + 3)*x + (n^3 + 6*n^2 + 8*n + 3)*x^2 + (n^3 + 3*n^2 + 3*n + 1)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*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] = ((n + 1 + (3*n^2 + 6*n + 3)*x + (n^3 + 6*n^2 + 8*n + 3)*x^2 + (n^3 + 3*n^2 + 3*n + 1)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*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.