Real-rooted atlas

A038284 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 720\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04007203241444
180002160019440583254872
21600005760007776004665601049762085136
3320000014400000259200002332800010497600188956879235168
464000000345600000777600000933120000629856000226748160340122243010936384
512800000008064000000217728000003265920000029393280000158723712004761711360612220032114415582592
6256000000001843200000005806080000001045094400000117573120000084652646400038093690880097955205120110199605764347792138496
75120000000004147200000000149299200000003135283200000042326323200000380936908800002285621452800088159684608001983592903680198359290368165216101262848
810240000000000921600000000003732480000000008957952000000001410877440000000152374763520000011428107264000005877312307200001983592903680003967185807360035704672266246278211847988224
920480000000000020275200000000009123840000000000246343680000000004434186240000000055870746624000000502836719616000003232521768960000014546347960320000436390438809600078550278985728064268410079232238572050223552512
104096000000000000442368000000000002189721600000000006569164800000000001330255872000000000191556845568000000020113468784640000001551610449100800000872780877619200000349112351047680000942603347828736001542441841901568011568313814261769065737908494995456
11819200000000000009584640000000000005175705600000000000170798284800000000003842961408000000000062255974809600000000747071697715200000006723645279436800000045384605636198400000226923028180992000008169229014515712000200517439447203840030077615917080576020822964865671168344498040522809827328
1216384000000000000002064384000000000000012076646400000000000043475927040000000000010760291942400000000001936852549632000000000261475094200320000000026894581117747200000002117948263022592000000127076895781355520000057184603101609984000018714961015072358400042108662283912806400583043016238792704037481336758208102413090925539866773438464
13327680000000000000004423680000000000000002786918400000000000000108689817600000000000002934625075200000000000058105576488960000000000871583647334400000000001008546791915520000000009076921127239680000000063538447890677760000000343107618609659904000001403622076130426880000042108662283912806400008745645243581890560001124440102746243072006746640616477458432497455170514937390661632
146553600000000000000009437184000000000000000637009920000000000000002675441664000000000000007825666867200000000000001690344043315200000000000278906767147008000000000035859441490329600000000003630768450895872000000000290461476071669760000000018299072992515194880000008983181287234732032000003368692982713024512000009328688259820683264000017991041643939889152000215892499727278669824012143953109659425177618903296479567620845142016
Technical details
Node id
n-24957611ccdbbb19 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-249d425e0def4c4f

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:             400,              720,              324;
  1:            8000,            21600,            19440,              5832;
  2:          160000,           576000,           777600,            466560,            104976;
  3:         3200000,         14400000,         25920000,          23328000,          10497600,           1889568;
  4:        64000000,        345600000,        777600000,         933120000,         629856000,         226748160,          34012224;
  5:      1280000000,       8064000000,      21772800000,       32659200000,       29393280000,       15872371200,        4761711360,         612220032;
  6:     25600000000,     184320000000,     580608000000,     1045094400000,     1175731200000,      846526464000,      380936908800,       97955205120,       11019960576;
  7:    512000000000,    4147200000000,   14929920000000,    31352832000000,    42326323200000,    38093690880000,    22856214528000,     8815968460800,     1983592903680,     198359290368;
  8:  10240000000000,   92160000000000,  373248000000000,   895795200000000,  1410877440000000,  1523747635200000,  1142810726400000,   587731230720000,   198359290368000,   39671858073600,   3570467226624;
  9: 204800000000000, 2027520000000000, 9123840000000000, 24634368000000000, 44341862400000000, 55870746624000000, 50283671961600000, 32325217689600000, 14546347960320000, 4363904388096000, 785502789857280, 64268410079232;
%F Formula
The row polynomials satisfy P(n,x) = (20+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 720*x + 324*x^2.
%t Mathematica
P[0, x_] := 400 + 720*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 18*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 720*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 18*x)*v[i - 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.