Real-rooted atlas

A038284 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(182 + 120\,x + 18\,x^{2} + 42\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 364 + 240\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
036424036640
162726552216021615200
210427215052878624172801296352000
316903043128160225792078624012960077768000000
426891200608509445630688027095040707616093312046656179200000
5421654016112943040012778698247882963202844979205943974465318402799363968000000
665356372482023939276827106329600204459171849459555840273118003247551795244789760167961687040000000
7100353655808352924411392546463604736487913932800276019881984102163203072245806202883668281344302330880100776961894400000000
81528917462016602121934848010587732341760109292720947207318708992000331223858380810216320307202106910310402751211008020155392006046617640960000000000
92314037239808010090855249305619870023849984023293011151872018033298956288096606958694400364346244218889632530575360173820100608020175547392013302558720362797056880640000000000
103482626045911041666106812661760363270788975001647688057239961604192742007336960259679504970547211592835043328003747561369108488669277517824013905608048640145263941222487071293440217678233618841600000000000
1152158399385272322716448315810611264978165693808640944504051335004169299171161792512065406775314456576337583356461711361291773047685120036538723348807687513373848780801084637427793921030053401395256596340736013060694016401408000000000000
12777840477789061124381305548362874881140908292640456704181938863942664192019834585078035087361562260755181142016915694854402392064405100027754053632135636170006937600341028084588871686311234032975872828268581224448721037380976643656994324480783641640968519680000000000000
13115564870985803366470005643001015500801971587496763293696034227248779213701120409362443870994432003570225314046315724823433911327717130240117732195566021836804557375312233103360135636170006937600030692527612998451251637369360711680621201435918336049917972529152023509249228800470184984576180224000000000000000
14171124905113593446401109422761463712317443360270864048744038406309079989642539827208214539707011288268807859758922323093094405712360502474105159683213793553515492147201412786346792262041604861200333048643584013021072320666009600267862059167986483241309895488569344045873336806277120342294668771328015045919506432028211099074563801088000000000000000
Technical details
Node id
n-547ce693e7bbd987 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3150a1751f9b8c12

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:            364,             240,              36;
  1:           6272,            6552,            2160,             216;
  2:         104272,          150528,           78624,           17280,            1296;
  3:        1690304,         3128160,         2257920,          786240,          129600,           7776;
  4:       26891200,        60850944,        56306880,        27095040,         7076160,         933120,          46656;
  5:      421654016,      1129430400,      1277869824,       788296320,       284497920,       59439744,        6531840,        279936;
  6:     6535637248,     20239392768,     27106329600,     20445917184,      9459555840,     2731180032,      475517952,      44789760,       1679616;
  7:   100353655808,    352924411392,    546463604736,    487913932800,    276019881984,   102163203072,    24580620288,    3668281344,     302330880,     10077696;
  8:  1528917462016,   6021219348480,  10587732341760,  10929272094720,   7318708992000,  3312238583808,  1021632030720,  210691031040,   27512110080,   2015539200,    60466176;
  9: 23140372398080, 100908552493056, 198700238499840, 232930111518720, 180332989562880, 96606958694400, 36434624421888, 9632530575360, 1738201006080, 201755473920, 13302558720, 362797056;
%F Formula
The row polynomials satisfy (3*n+10+3*x)*P(n,x) = (42*n+182+(18*n+120)*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 364 + 240*x + 36*x^2.
%t Mathematica
P[0, x_] := 364 + 240*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((42*n + 182 + (18*n + 120)*x + 18*x^2)*P[n - 1, x])/(3*n + 10 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 364 + 240*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((42*n + 182 + (18*n + 120)*x + 18*x^2)*v[i - 1])/(3*n + 10 + 3*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. [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.