Real-rooted atlas

A038334 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 100\,x + 64\,x^{2} + 60\,n x + 64\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(500 + 3600\,x + 5440\,x^{2} + 2304\,x^{3} + 1200\,n x + 2880\,n x^{2} + 1536\,n x^{3} + 320\,n^{2} x^{2} + 256\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 2560\,x + 2304\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0400256023045264
180007680013824065536288576
2160000204800055296005242880163840014618880
332000005120000018432000026214400016384000037748736702452736
46400000012288000005529600000104857600009830400000452984832082208358432490491904
5128000000028672000000154828800000367001600000458752000000317089382400115091701760171798691841459895353344
62560000000065536000000041287680000001174405120000018350080000000169114337280009207336140800274877906944034789235097664119300489216
7512000000000147456000000001061683200000003523215360000006606028800000007610145177600005524401684480002473901162496006262062317568068719476736002764687709306880
8102400000000003276800000000002654208000000000100663296000000002202009600000000030440580710400000276220084224000001649267441664000062620623175680001374389534720000133040906960896117403309908688896
9204800000000000720896000000000064880640000000000276824064000000000692060160000000000111615462604800000012153683705856000009070970929152000004592179032883200001511828488192000002926899953139712025332747903959044922001739978113024
10409600000000000015728640000000000015571353600000000007381975040000000000207618048000000000003826815860736000000048614734823424000000435406604599296000002755307419729920000012094627905536000000351227994376765440060798594969501696047569271064100864204101388758075572224
1181920000000000000340787200000000000036805017600000000000191931351040000000000599785472000000000000124371515473920000000018056901505843200000001886761953263616000000143275985825955840000078615081385984000000030439759512653004800079038173460352204800123680104766662246408827055269646172168383776047937047494656
121638400000000000000734003200000000000008587837440000000000004885525299200000000000167939932160000000000003869336036966400000000063199155270451200000000754704781305446400000006686212671877939200000044024445576151040000000213078316588571033600007376896189632872448000173152146673327144960024715754755009282048016212958658533785600341542526866222145863680
133276800000000000000015728640000000000000001981808640000000000000012213813248000000000000045801799680000000000000011608008110899200000000002106638509015040000000000283014292989542400000000028655197165191168000000002201222278807552000000000127846989953142620160000055326721422246543360000017315214667332714496000037073632132513923072000486388759756013568000029514790517935282585613813026021069524991737856
146553600000000000000003355443200000000000000045298483200000000000000030064771072000000000000001221381324800000000000000033768750868070400000000000674124322884812800000000001006273041740595200000000001146207886607646720000000001006273041740595200000000006818506130834273075200000035409101710237787750400000138521717338661715968000003954520760801485127680000778222015609621708800000944473296573929042739205331109037302060417024555042921112187883900370944
Technical details
Node id
n-d85cea0fb7505d23 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7f34919e5b06c424

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:             400,             2560,              2304;
  1:            8000,            76800,            138240,              65536;
  2:          160000,          2048000,           5529600,            5242880,            1638400;
  3:         3200000,         51200000,         184320000,          262144000,          163840000,            37748736;
  4:        64000000,       1228800000,        5529600000,        10485760000,         9830400000,          4529848320,           822083584;
  5:      1280000000,      28672000000,      154828800000,       367001600000,       458752000000,        317089382400,        115091701760,        17179869184;
  6:     25600000000,     655360000000,     4128768000000,     11744051200000,     18350080000000,      16911433728000,       9207336140800,      2748779069440,       347892350976;
  7:    512000000000,   14745600000000,   106168320000000,    352321536000000,    660602880000000,     761014517760000,     552440168448000,    247390116249600,     62620623175680,      6871947673600;
  8:  10240000000000,  327680000000000,  2654208000000000,  10066329600000000,  22020096000000000,   30440580710400000,   27622008422400000,  16492674416640000,   6262062317568000,   1374389534720000,   133040906960896;
  9: 204800000000000, 7208960000000000, 64880640000000000, 276824064000000000, 692060160000000000, 1116154626048000000, 1215368370585600000, 907097092915200000, 459217903288320000, 151182848819200000, 29268999531397120, 2533274790395904;
%F Formula
The row polynomials satisfy (25+(60*n+100)*x+(16*n^2+64*n+64)*x^2)*P(n,x) = (500+(1200*n+3600)*x+(320*n^2+2880*n+5440)*x^2+(256*n^2+1536*n+2304)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 2560*x + 2304*x^2.
%t Mathematica
P[0, x_] := 400 + 2560*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((500 + (1200*n + 3600)*x + (320*n^2 + 2880*n + 5440)*x^2 + (256*n^2 + 1536*n + 2304)*x^3)*P[n - 1, x])/(25 + (60*n + 100)*x + (16*n^2 + 64*n + 64)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 2560*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((500 + (1200*n + 3600)*x + (320*n^2 + 2880*n + 5440)*x^2 + (256*n^2 + 1536*n + 2304)*x^3)*v[i - 1])/(25 + (60*n + 100)*x + (16*n^2 + 64*n + 64)*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.