Real-rooted atlas

A145324 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 6\,n x - 3\,n^{2} - 12\,n^{2} x - 9\,n^{2} x^{2} - n^{3} - 6\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,n + 6\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
26315954276
3459216328266486096
436542584862631559989720157851
5320583060001106667182746812509641749604698117
63093843741516181821874463880356283984313440843674160158174118
732682964804023629486357196977004318128078641856091348873505296881798405946526494

All 15 rows

Technical details
Node id
n-8b3634df3084005c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e2fa454bbad365d7

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:         9,          6;
  2:        63,        159,          54;
  3:       459,       2163,        2826,          648;
  4:      3654,      25848,       62631,        55998,          9720;
  5:     32058,     306000,     1106667,      1827468,       1250964,        174960;
  6:    309384,    3741516,    18182187,     44638803,      56283984,      31344084,       3674160;
  7:   3268296,   48040236,   294863571,    969770043,    1812807864,    1856091348,     873505296,      88179840;
  8:  37581840,  651991464,  4852592712,  20110498776,   50244947109,   76109272812,   65768172588,   26847297072,   2380855680;
  9: 467832240, 9372384504, 82181624112, 412238074896, 1295476920549, 2618126589042, 3341678417748, 2502378994392, 903033995040, 71425670400;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(6*n^3+12*n^2+6*n)*x+(9*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(6*n^2+6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*P[n - 1, x])/(n)]];
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] = 9 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*v[i - 1])/(n)); 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.