Real-rooted atlas

A038330 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 10\,x + 4\,x^{2} + 6\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(64 + 288\,x + 272\,x^{2} + 72\,x^{3} + 96\,n x + 144\,n x^{2} + 48\,n x^{3} + 16\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 1024\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
025610245761856
140962457627648819264512
2655365242888847365242881024002101248
310485761048576023592960209715208192000117964865470464
416777216201326592566231040671088640393216000113246208128450561974730752
526843545637580963841268357529618790481920146800640006341787648143864627213421772858095304704
6429496729668719476736270582939648481036337152469762048000270582939648920733614081717986918413589544961675590893568
7687194767361236950581248556627761561611544872091648135291469824009740985827328441952134758412369505812481956894474241342177280047552535724032
8109951162777621990232555520111325552312320263882790666240360777252864000311711546474496176780853903360659706976665601565515579392021474836480001299227607041331471000272896
91759218604441638702809297715221770330229964805805421394657280907097092915200091435386965852166222686057398272290271069732864091843580657664018897856102400022866405883904123695058124836858498801205248
1028147497671065667553994410557444179903404153241612384898975268864021770330229964800025079420424919449619912595383674470411146409077741977644084918715678720120946279055360002195174964854784237494511599616116135915683841010396279064231936
11450359962737049611709359031163289679038173460352204825760589868559237125031365208702976000652064931047905689659168854854346997763864088480283885568183393261857223475262892065108787200015219879756326502424699429206360064241562704622387210775213952204827463301271914545152
12720575940379279362017612633061982208147537923792657448965245792845961153740811270258067494666240016229171617192319385616567279359217159372812365083136908433817668466817760030097408281756451687366656008523132663542841344184422404740821811227055022917707366424136479252938752989560464998400740924808782715813888
131152921504606846976345876451382054092802723777054633675980801049158569192230748160245896539654429081600038950011881261566525444417941162457909166080370952494107253014528023474337517724604825601127025806749466624000409110367850056384512110653442844493086720216440183341658931202896377510352650240237494511599616000900719925474099219857719796271146860544
141844674407370955161659029581035870565171249806208999015789363202066035336255469780992052457928459611537408000906473003782087366410241130992937589224746516481055153761016186352435207511788005671873544192041216943789694779392000174553756949357390725125665456273638046040064138521717338661715968024715754755009282048030399297484750848000230584300921369395281346268269379584529090432537936319741952
Technical details
Node id
n-50e7a18aff9ba91f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            256,            1024,              576;
  1:           4096,           24576,            27648,             8192;
  2:          65536,          524288,           884736,           524288,           102400;
  3:        1048576,        10485760,         23592960,         20971520,          8192000,          1179648;
  4:       16777216,       201326592,        566231040,        671088640,        393216000,        113246208,         12845056;
  5:      268435456,      3758096384,      12683575296,      18790481920,      14680064000,       6341787648,       1438646272,        134217728;
  6:     4294967296,     68719476736,     270582939648,     481036337152,     469762048000,     270582939648,      92073361408,      17179869184,      1358954496;
  7:    68719476736,   1236950581248,    5566277615616,   11544872091648,   13529146982400,    9740985827328,    4419521347584,    1236950581248,    195689447424,     13421772800;
  8:  1099511627776,  21990232555520,  111325552312320,  263882790666240,  360777252864000,  311711546474496,  176780853903360,   65970697666560,  15655155793920,   2147483648000,   129922760704;
  9: 17592186044416, 387028092977152, 2177033022996480, 5805421394657280, 9070970929152000, 9143538696585216, 6222686057398272, 2902710697328640, 918435806576640, 188978561024000, 22866405883904, 1236950581248;
%F Formula
The row polynomials satisfy (4+(6*n+10)*x+(n^2+4*n+4)*x^2)*P(n,x) = (64+(96*n+288)*x+(16*n^2+144*n+272)*x^2+(8*n^2+48*n+72)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 1024*x + 576*x^2.
%t Mathematica
P[0, x_] := 256 + 1024*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((64 + (96*n + 288)*x + (16*n^2 + 144*n + 272)*x^2 + (8*n^2 + 48*n + 72)*x^3)*P[n - 1, x])/(4 + (6*n + 10)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 1024*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((64 + (96*n + 288)*x + (16*n^2 + 144*n + 272)*x^2 + (8*n^2 + 48*n + 72)*x^3)*v[i - 1])/(4 + (6*n + 10)*x + (n^2 + 4*n + 4)*x^2)); 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.