Real-rooted atlas

A038320 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(41 + 24\,x + 24\,n\right)\,P_{n}(x) = \left(1105 + 1968\,x + 576\,x^{2} + 408\,n + 576\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1105 + 1968\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0110519685763649
125721795607084813824189953
25551692469216381888022671363317769442177
31144237766620280148152960152755200680140807962624454947521
42285969771647702288479666016071113420805499187200195880550419110297621433397185
5446545026538404292136138406992192268612968960298676367360184772689920548465541124586471424992771786369
6857607826578573664508803686812045056885804750028812893422510080114691725066245912726077440150436262707211007531417645377745814273
7162535148375318524329053912925955766950402654504672440324783345650155525569958524354564128902102384641824498332467204061779093094426418075402242052125783884097

All 15 rows

Technical details
Node id
n-b7de04a38e112919 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b561c2303414350e

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1105,             1968,               576;
  1:           25721,            79560,             70848,              13824;
  2:          555169,          2469216,           3818880,            2267136,             331776;
  3:        11442377,         66620280,         148152960,          152755200,           68014080,            7962624;
  4:       228596977,       1647702288,        4796660160,         7111342080,         5499187200,         1958805504,           191102976;
  5:      4465450265,      38404292136,      138406992192,       268612968960,       298676367360,       184772689920,         54846554112,         4586471424;
  6:     85760782657,     857366450880,     3686812045056,      8858047500288,     12893422510080,     11469172506624,       5912726077440,      1504362627072,       110075314176;
  7:   1625351483753,   18524329053912,    92595576695040,    265450467244032,    478334565015552,    556995852435456,     412890210238464,    182449833246720,     40617790930944,      2641807540224;
  8:  30477084259729,  390084356100720,  2222919486469440,   7407646135603200,  15927028034641920,  22960059120746496,   22279834097418240,  14156235779604480,   5473494997401600,   1083141091491840,    63403380965376;
  9: 566494286026169, 8045950244568456, 51491135005295040, 195616914809310720, 488904644949811200, 840947080229093376, 1010242601312845824, 840268028816916480, 467155780726947840, 160555853257113600, 28594924815384576, 1521681143169024;
%F Formula
The row polynomials satisfy (24*n+41+24*x)*P(n,x) = (408*n+1105+(576*n+1968)*x+576*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1105 + 1968*x + 576*x^2.
%t Mathematica
P[0, x_] := 1105 + 1968*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((408*n + 1105 + (576*n + 1968)*x + 576*x^2)*P[n - 1, x])/(24*n + 41 + 24*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1105 + 1968*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((408*n + 1105 + (576*n + 1968)*x + 576*x^2)*v[i - 1])/(24*n + 41 + 24*x)); 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.