Real-rooted atlas

A225470 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 24\,x - 16\,x^{2} + 18\,n - 24\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 8\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2363268
33965761921164
45400100805760102422264
588776194400146880460805120481256
617146084209408381024016128003225602457611694192
73810628810211443210527148854190080148377602064384114688316699120
8958640832275559667231392023041862009856627056640120766464123863045242889476183360
9269414605448207023680010129204224067016678400262405785606270566400898007040707788802359296310802708160
1083681408448026776387883523531182688000255072457728011223007488003140717690885620285440062285414403892838401048576011095563821440
1128471972832640950565926914561325948614341121030068853862404978954759680015751411974144332036245094446246920192040874803200207618048046137344428497100689280
12105327558745344036498131195535365343754567776768441542046821990423074581404160008060406595338241930396374466563182769104486435526043238402564082892801079613849620132659217804449881523200
13420959922679065601507689163622154242303053068606151682007014226220523521120679698173235204251138479581593611290068938244096212374686479155228177617833164825789275832320154924587417654962159616872415232792173979063577600
1418077432051837491206669062425356300288105774363863191296009657754720049479680571036184543784960023234044165474222086725693577867264001408091120389324802139146038149120023354881553203201785411403776009068756336640274810798080375809638437583056305722009600
Technical details
Node id
n-2e43a111a868ac2c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ea29b0775a168846

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           0;
  1:           4;
  2:          36,          32;
  3:         396,         576,          192;
  4:        5400,       10080,         5760,        1024;
  5:       88776,      194400,       146880,       46080,        5120;
  6:     1714608,     4209408,      3810240,     1612800,      322560,      24576;
  7:    38106288,   102114432,    105271488,    54190080,    14837760,    2064384,    114688;
  8:   958640832,  2755596672,   3139202304,  1862009856,   627056640,  120766464,  12386304,   524288;
  9: 26941460544, 82070236800, 101292042240, 67016678400, 26240578560, 6270566400, 898007040, 70778880, 2359296;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(24*n-24)*x+16*x^2)*P(n-2,x) + (6*n-3+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (24*n - 24)*x + 16*x^2)*P[n - 2, x] + (6*n - 3 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (24*n - 24)*x + 16*x^2)*v[i - 2] + (6*n - 3 + 8*x)*v[i - 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.