Real-rooted atlas

A038284 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 14\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 672\,x + 196\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
05766721961444
1138242419214112274454872
2331776774144677376263424384162085136
37962624232243202709504015805440460992053782479235168
41911029766688604169754214407586611203319142407744665675295363010936384
545864714241872809164832774160384318637670401858719744065055191041264962048105413504114415582592
6110075314176513684799488104877313228812235686543368921854771204163532226561214363566082023939276814757890564347792138496
7264180754022413869489586176323621423677444404847155609638542412611584224830740234248743417675776218585441894431877043609620661046784165216101262848

All 15 rows

Technical details
Node id
n-76d494d4a790974a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-48a0cbb267bcb941

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:              576,              672,               196;
  1:            13824,            24192,             14112,              2744;
  2:           331776,           774144,            677376,            263424,             38416;
  3:          7962624,         23224320,          27095040,          15805440,           4609920,            537824;
  4:        191102976,        668860416,         975421440,         758661120,         331914240,          77446656,           7529536;
  5:       4586471424,      18728091648,       32774160384,       31863767040,       18587197440,        6505519104,        1264962048,         105413504;
  6:     110075314176,     513684799488,     1048773132288,     1223568654336,      892185477120,      416353222656,      121436356608,       20239392768,       1475789056;
  7:    2641807540224,   13869489586176,    32362142367744,    44048471556096,    38542412611584,    22483074023424,     8743417675776,     2185854418944,     318770436096,     20661046784;
  8:   63403380965376,  369853055631360,   970864271032320,  1510233310494720,  1541696504463360,  1079187553124352,   524605060546560,   174868353515520,   38252452331520,   4958651228160,   289254654976;
  9: 1521681143169024, 9764120668667904, 28478685283614720, 49837699246325760, 58143982454046720, 47484252337471488, 27699147196858368, 11541311332024320, 3366215805173760, 654541962117120, 76363228913664, 4049565169664;
%F Formula
The row polynomials satisfy P(n,x) = (24+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 672*x + 196*x^2.
%t Mathematica
P[0, x_] := 576 + 672*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 14*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] = 576 + 672*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 14*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

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.