Real-rooted atlas

A034870 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-3 + 2\,x + 4\,n\right)\,P_{n}(x) = \left(4 + 24\,x + 48\,x^{2} + 32\,x^{3} + 16\,n + 64\,n x + 64\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
120481684
2144896192015362564752
38328448345607168076800368644096233280
443526144037273612615682580480321126422937607864326553610637568
521504389120313344014745600447283209083289612386304011010048058982400157286401048576463574016
61024002260992227082241369702405514854401557135360314887372845675970564671406080322961408013841203203019898881677721619591041024
74751361238630414909440010973347845510529024199428669445353085337610796138496016374143385618473392537615114593894485496692736305345331205637144576268435456809763028992
82162688650117129122611207927234560477102080002106882785287053477150721823347834880367068708864057579405312006986301177856644889339494443969727692802104533975040644245094400103079215104429496729632912948920320
9969932833030144052942602245304955699237220253696019405870202887786305945600245602336112646166754308915212398765277184019968537762201625628069855232025912826186956820239603885670411828339933184049065706389504131425999257601855425871872687194767361320279436689408
10429916161635778560294859571203347054592002682425180160161270653255687543304749056028092307341312084537035980800020739752827289604165985133133824685313571684352091976295147110409988375943577600865659248443392058598472202321922996512783073280109676284870656026113401159680032985348833280109951162777652404937640902656
11188743680793562316815889701273620152791859201816334958592012370713772032066094956439142428383634972999689957162268753920288414355370803206943308555223040013937113899938611223329516745549414432473320012120064037356526028980224035159083321393152026662304852056473615924996563219251272386348014632960237054706948505605079743720325120580542139465728175921860444162062073360195518464
12822083584378493992968334920908801168123730329611694820137369688994728850227253472901180948482601384381775872010424118844116172834817258900212940897712952397371801623157050787590307844645704633312870400789055063873447526411321224829488594944136573505879544954881374720157866472243211416361172593541120769547308671120179241125605159752171521682411120171679744498492183754571776971088669651763201013309916158361628147497671065680552482751467487232
13355676979217797270732842748346368006560562544640072235981209600060742433977139204054204314288128022036790001008640099306816478904320037575552181207040001204564844208979968032918135687474053120769859624948989952001543803385839353856002654688361893068800003907701268706597273604905864092778391142405221928165982601216004672251516931801088003471641994005053440002106129476363065753601018348318241482342403787245811641876480010245689152267878400182958734861926400017564038546744934445035996273704963127331683292267151360
14153008209928267812044802151134370201635879297797324843090333138944003967828226998272029125547623710720017493820985416089608757081318862356480370218072016945152001334683613476474060804131944356217094144001103819420875138007040255242159792106504192051166055073532870656008892307502434678210560133796293439410667520001737777740671875022848019392592178512228515840184691354081068843008001487251430231764893696099983289427345539072005514229901750572154880243437574257884790784082990082133369815040020655309330972042854434047213182920949760302641894959297331272057594037927936120771863551141371445248
Technical details
Node id
n-de82162bebd38fff · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ec29035d385d8da2

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:       1;
  1:      20,        48,         16;
  2:     144,       896,       1920,        1536,          256;
  3:     832,      8448,      34560,       71680,        76800,         36864,          4096;
  4:    4352,     61440,     372736,     1261568,      2580480,       3211264,       2293760,         786432,          65536;
  5:   21504,    389120,    3133440,    14745600,     44728320,      90832896,     123863040,      110100480,       58982400,        15728640,         1048576;
  6:  102400,   2260992,   22708224,   136970240,    551485440,    1557135360,    3148873728,     4567597056,     4671406080,      3229614080,      1384120320,       301989888,        16777216;
  7:  475136,  12386304,  149094400,  1097334784,   5510529024,   19942866944,   53530853376,   107961384960,   163741433856,    184733925376,    151145938944,     85496692736,     30534533120,      5637144576,       268435456;
  8: 2162688,  65011712,  912261120,  7927234560,  47710208000,  210688278528,  705347715072,  1823347834880,  3670687088640,   5757940531200,   6986301177856,   6448893394944,   4396972769280,   2104533975040,    644245094400,   103079215104,     4294967296;
  9: 9699328, 330301440, 5294260224, 53049556992, 372202536960, 1940587020288, 7786305945600, 24560233611264, 61667543089152, 123987652771840, 199685377622016, 256280698552320, 259128261869568, 202396038856704, 118283399331840, 49065706389504, 13142599925760, 1855425871872, 68719476736;
%F Formula
The row polynomials satisfy (4*n-3+2*x)*P(n,x) = (16*n+4+(64*n+24)*x+(64*n+48)*x^2+32*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 4 + (64*n + 24)*x + (64*n + 48)*x^2 + 32*x^3)*P[n - 1, x])/(4*n - 3 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 4 + (64*n + 24)*x + (64*n + 48)*x^2 + 32*x^3)*v[i - 1])/(4*n - 3 + 2*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.