Real-rooted atlas

A038334 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 960\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
05769604001936
1138243456028800800085184
2331776110592013824007680001600003748096
37962624331776005529600046080000192000003200000164916224
4191102976955514880199065600022118400001382400000460800000640000007256313856
545864714242675441664066886041600928972800007741440000038707200000107520000001280000000319277809664
6110075314176733835427840214035333120035672555520003715891200000247726080000010321920000002457600000002560000000014048223625216
72641807540224198135565516806604518850560012842119987200016052649984000013377208320000074317824000000265420800000005529600000000512000000000618121839509504

All 15 rows

Technical details
Node id
n-740344b12e5a271f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-da2debf48cb6165b

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,               960,               400;
  1:            13824,             34560,             28800,               8000;
  2:           331776,           1105920,           1382400,             768000,             160000;
  3:          7962624,          33177600,          55296000,           46080000,           19200000,            3200000;
  4:        191102976,         955514880,        1990656000,         2211840000,         1382400000,          460800000,           64000000;
  5:       4586471424,       26754416640,       66886041600,        92897280000,        77414400000,        38707200000,        10752000000,         1280000000;
  6:     110075314176,      733835427840,     2140353331200,      3567255552000,      3715891200000,      2477260800000,      1032192000000,       245760000000,       25600000000;
  7:    2641807540224,    19813556551680,    66045188505600,    128421199872000,    160526499840000,    133772083200000,     74317824000000,     26542080000000,     5529600000000,      512000000000;
  8:   63403380965376,   528361508044800,  1981355655168000,   4403012567040000,   6421059993600000,   6421059993600000,   4459069440000000,   2123366400000000,   663552000000000,   122880000000000,   10240000000000;
  9: 1521681143169024, 13948743812382720, 58119765884928000, 145299414712320000, 242165691187200000, 282526639718400000, 235438866432000000, 140142182400000000, 58392576000000000, 16220160000000000, 2703360000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = (24+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 960*x + 400*x^2.
%t Mathematica
P[0, x_] := 576 + 960*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 20*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 + 960*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 20*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.