Real-rooted atlas

A038334 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(20 + 56\,x + 36\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1440\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0400144012963136
18000432007776046656175616
216000011520003110400373248016796169834496
332000002880000010368000018662400016796160060466176550731776
46400000069120000031104000007464960000100776960007255941120217678233630840979456
512800000001612800000087091200000261273600000470292480000507915878400304749527040783641640961727094849536
6256000000003686400000002322432000000836075520000018811699200000270888468480002437996216320012538266255360282110990745696717311574016
75120000000008294400000000597196800000002508226560000006772211712000001218998108160000146279772979200011284439629824005077997833420801015599566684165416169448144896
810240000000000184320000000000149299200000000071663616000000002257403904000000048759924326400000731398864896000007522959753216000050779978334208000203119913336832003656158440062976303305489096114176
92048000000000004055040000000000364953600000000001970749440000000007094697984000000001787863891968000000321815500554240000041376278642688000003723865077841920000223431904670515200080435485681385472013162170384226713616985107389382393856
104096000000000000884736000000000008758886400000000005255331840000000000212840939520000000006129819058176000000012872620022169600000019860613748490240000022343190467051520000017874552373641216000096522582817662566400315892089221441126404738381338321616896951166013805414055936
1181920000000000000191692800000000000020702822400000000000136638627840000000000614873825280000000000199219119390720000000047812588653772800000008606265957679104000000116184590428667904000001161845904286679040000083652905108640890880004106597159878734643200123197914796362039296017058172817957820825653265296773103187132416
12163840000000000000041287680000000000000483065856000000000000347807416320000000000017216467107840000000000619792815882240000000001673440602882048000000003442506383071641600000005421947553337835520000006506337064005402624000005855703357604862361600003832824015886819000320001724770807149068550144004776288389028189831168061409422144648154972162982856619293778479415296
133276800000000000000088473600000000000000011147673600000000000000869518540800000000000004695400120320000000000001859378447646720000000000557813534294016000000000012909398936518656000000000232369180857335808000000003253168532002701312000000035134220145629174169600000287461801191511425024000001724770807149068550144000071644325835422847467520001842282664339444649164800221073919720733357899776167039970680451594847256576
14655360000000000000000188743680000000000000002548039680000000000000002140353331200000000000000125210669875200000000000005409100938608640000000000017850033097408512000000000045900085107621888000000000092947672342934323200000000014871627574869491712000000001873825074433555955712000000183975552762567312015360000013798166457192548401152000007642061422445103729868800002947652262943111438663680007074365431063467452792832079586611099464008843919369354238358105289311446368256
Technical details
Node id
n-352735e5f7822499 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-dca555f310728b4d

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,             1440,              1296;
  1:            8000,            43200,             77760,              46656;
  2:          160000,          1152000,           3110400,            3732480,            1679616;
  3:         3200000,         28800000,         103680000,          186624000,          167961600,            60466176;
  4:        64000000,        691200000,        3110400000,         7464960000,        10077696000,          7255941120,          2176782336;
  5:      1280000000,      16128000000,       87091200000,       261273600000,       470292480000,        507915878400,        304749527040,         78364164096;
  6:     25600000000,     368640000000,     2322432000000,      8360755200000,     18811699200000,      27088846848000,      24379962163200,      12538266255360,       2821109907456;
  7:    512000000000,    8294400000000,    59719680000000,    250822656000000,    677221171200000,    1218998108160000,    1462797729792000,    1128443962982400,     507799783342080,     101559956668416;
  8:  10240000000000,  184320000000000,  1492992000000000,   7166361600000000,  22574039040000000,   48759924326400000,   73139886489600000,   75229597532160000,   50779978334208000,   20311991333683200,   3656158440062976;
  9: 204800000000000, 4055040000000000, 36495360000000000, 197074944000000000, 709469798400000000, 1787863891968000000, 3218155005542400000, 4137627864268800000, 3723865077841920000, 2234319046705152000, 804354856813854720, 131621703842267136;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (20+56*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1440*x + 1296*x^2.
%t Mathematica
P[0, x_] := 400 + 1440*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((20 + 56*x + 36*x^2)*P[n - 1, x])/(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 + 1440*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((20 + 56*x + 36*x^2)*v[i - 1])/(1 + 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.