Real-rooted atlas

A038320 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(34 + 36\,x + 17\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 612 + 648\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
06126481260
115606330481749666150
2353736112363211897284199043087000
375168903183624050563440356918409447840135056250
4153344556811824120171915696018202838409636796802040733445672362500
5304133369419321414056511449195607220459232057338940960242847279364285540224231621468750
6590887689124379520519361391141812032245495613888025993653235201651361499648582833470464881596846089264858750000
7113007270544295723805637443547411620681675121657849728994257236246408421943648204844586760490496134884317450241785233613312364803813281250
821345817769460203413086979560861514250736960212844697240896033804746032377603579326050487040252658309446144011465166983270403034897142630403570467226624014186814960937500
93991667922889024226471918353080201378956109764405685994054863936010535812513424352013386679428821529611811775966607232071466207529052160283762882835942406676773713786880706952510871552546192375996093750
107402729602448728862200271344028324564589671821326401449928483990303680307043678962652544045514710057993200644819204594375750656364477658398166016019295876032844083206810309188062617601442183122177966081388197657711411220854617992578125000
11136333603511764074173223872697300235210087743174725131344356037994402063459208482081631343276528014369644175452139059217750736922617348249616109912501198937907210660971508146355968050169277685394616320159361235000665251843067916823542218752270698543253725184790737598885253906250
12249595366429229612434356068084964546648218262079598598296352847370426676911032896224303936473299979296042749691421970113701126035250553689898404864639026529214224536985650746224378776654407682985072022280979671040126426579767194433126436508210200152403148864426253294386593792524738714614913433629804724881059570312500
1345462013171038250830673907489358919953480463806919147021379748019643587163873846671680571975038006914947204801211241256955819888198401923736113988655116550402327882356423246527590402156714536098007812326401522386731363299632230408059694460158645111808031031978670129542676480821434729503429070848013380837222680292556801011996092471618764801117677183039733886718750
1482437783883482694838413093059793259016239040970426784676844733011204452546423811405245580801414338275798916960360960329457621891983009589964858139580333879354633523207914800011839038193807360838037648312368749932544069014865155136249994444804384473786326302940823552211017454956880890200064074476748808310902423552018197938622845197877248027526293715228030402560194303249754550802841641726614833483398437500000
Technical details
Node id
n-0c0e7ad183455578 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a7a3fcc1bd498552

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:             612,              648;
  1:           15606,            33048,             17496;
  2:          353736,          1123632,           1189728,            419904;
  3:         7516890,         31836240,          50563440,          35691840,            9447840;
  4:       153344556,        811824120,        1719156960,        1820283840,          963679680,          204073344;
  5:      3041333694,      19321414056,       51144919560,       72204592320,        57338940960,        24284727936,         4285540224;
  6:     59088768912,     437952051936,     1391141812032,     2454956138880,      2599365323520,      1651361499648,       582833470464,       88159684608;
  7:   1130072705442,    9572380563744,    35474116206816,    75121657849728,     99425723624640,     84219436482048,     44586760490496,    13488431745024,     1785233613312;
  8:  21345817769460,  203413086979560,   861514250736960,  2128446972408960,   3380474603237760,   3579326050487040,   2526583094461440,  1146516698327040,   303489714263040,   35704672266240;
  9: 399166792288902, 4226471918353080, 20137895610976440, 56859940548639360, 105358125134243520, 133866794288215296, 118117759666072320, 71466207529052160, 28376288283594240, 6676773713786880, 706952510871552;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(18*n+36)*x)*P(n-1,x) for n >= 1, with P(0,x) = 612 + 648*x.
%t Mathematica
P[0, x_] := 612 + 648*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (18*n + 36)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 612 + 648*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (18*n + 36)*x)*v[i - 1])/(n + 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.