Real-rooted atlas

A038301 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(64 + 88\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 4096 + 11264\,x + 7744\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0409611264774423104
1262144108134414868486814723511808
2167772169227468819031654417445683259969536533794816
310737418247381975040203004313602791309312019190251520527731916881136812032
468719476736566935683072194884141056035728759193603684528291840202649056051246440408678412332795428864
5439804651110442331197669376174616190386176400162102968320550222891581440453933885554688208053030879232408675596369921874584905187328
628147497671065630962247438172161490058157962035240976599343955968704285301224243207747138313466675253261575905083392209241905341399043596345248055296284936905588473856
71801439850948198422292818155483955212261049985516175363933753537019772928811336667010328166411155879171392012288102262225737760112646026166873832292352207149486287985049631647838182886604843310409649448026112

All 15 rows

Technical details
Node id
n-8381edfecc6c9677 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-e348a50f4e54dede

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:                 4096,                  11264,                   7744;
  1:               262144,                1081344,                1486848,                  681472;
  2:             16777216,               92274688,              190316544,               174456832,                59969536;
  3:           1073741824,             7381975040,            20300431360,             27913093120,             19190251520,               5277319168;
  4:          68719476736,           566935683072,          1948841410560,           3572875919360,           3684528291840,            2026490560512,             464404086784;
  5:        4398046511104,         42331197669376,        174616190386176,         400162102968320,         550222891581440,          453933885554688,          208053030879232,           40867559636992;
  6:      281474976710656,       3096224743817216,      14900581579620352,       40976599343955968,       70428530122424320,        77471383134666752,        53261575905083392,        20924190534139904,         3596345248055296;
  7:    18014398509481984,     222928181554839552,    1226104998551617536,     3933753537019772928,     8113366670103281664,     11155879171392012288,     10226222573776011264,      6026166873832292352,      2071494862879850496,      316478381828866048;
  8:  1152921504606846976,   15852670688344145920,   98088399884129402880,   359657466241807810560,   865425778144350044160,   1427952533938177572864,   1636195611804161802240,   1285582266417555701760,    662878356121552158720,   202546164370474270720,    27850097600940212224;
  9: 73786976294838206464, 1116028016459427872768, 7672692613158566625280, 31649857029279087329280, 87037106830517490155520, 167546430648746168549376, 230376342142025981755392, 226262478889489803509760, 155555454236524239912960, 71296249858406943293440, 19606468711061909405696, 2450808588882738675712;
%F Formula
The row polynomials satisfy P(n,x) = (64+88*x)*P(n-1,x) for n >= 1, with P(0,x) = 4096 + 11264*x + 7744*x^2.
%t Mathematica
P[0, x_] := 4096 + 11264*x + 7744*x^2;
P[n_, x_] := P[n, x] = Expand[(64 + 88*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] = 4096 + 11264*x + 7744*x^2; for(n = 1, N, my(i = n + 1); v[i] = (64 + 88*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.