Real-rooted atlas

A085880 → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-30 - 72\,x + 20\,n + 48\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
251217
350240288578
4625450010800864024565
58750840003024004838402903041169294
61312501575000756000018144000217728001045094459633994
720625002970000017820000057024000010264320009853747203941498883186159108
83351562556306250040540500001621620000038918880000560431872004483454976015371845632176035290717
9558593750107250000009009000000043243200000012972960000002490808320000298896998400020495794176006148738252809975333140630
10949609375020511562500019691100000001102701600000039697257600000952734182400001524374691840001567928254464009407569526784025086852071424576574255528414
11164023437500393656250000042514875000000272095200000000114279984000000032912635392000006582527078400000902746570752000081247191367680004333183539609600103996404950630433860633551941404
12287041015625075778828125000909345937500000654729075000000031426995600000000105594705216000000253427292518400000434446787174400000521336144609280000417068915687424000200193079529963520436784900792647682014707696340513538
1350784179687500146258437500000019306113750000000154448910000000000834024114000000000320265259776000000089674272737280000001844727896309760000027670918444646400000295156463409561600002125126536548843520092732794322131353601854655886442627072121192416810637045132
1490686035156250028294042968750000407434218750000000358542112500000000021512526750000000000929341155600000000002973891697920000000007137340075008000000001284721213501440000000171296161800192000000016444431532818432000001076362791239024640000430545116495609856000794852522761125888007358111020645820597300
1516323486328125000548469140625000000855611859375000000082138738500000000000542115674100000000000260215523568000000000093677588484480000000002569442427002880000000053958290967060480000000863332655472967680000001035999186567561216000009041447446407806976000054248684678446841856000200302835735803723776003433762898328063836160450316394463524220554760
Technical details
Node id
n-d073c7e23a70d5a1 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f055f948fcf55643

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          1;
   2:          5,           12;
   3:         50,          240,           288;
   4:        625,         4500,         10800,           8640;
   5:       8750,        84000,        302400,         483840,         290304;
   6:     131250,      1575000,       7560000,       18144000,       21772800,       10450944;
   7:    2062500,     29700000,     178200000,      570240000,     1026432000,      985374720,       394149888;
   8:   33515625,    563062500,    4054050000,    16216200000,    38918880000,    56043187200,     44834549760,     15371845632;
   9:  558593750,  10725000000,   90090000000,   432432000000,  1297296000000,  2490808320000,   2988969984000,   2049579417600,   614873825280;
  10: 9496093750, 205115625000, 1969110000000, 11027016000000, 39697257600000, 95273418240000, 152437469184000, 156792825446400, 94075695267840, 25086852071424;
%F Formula
The row polynomials satisfy n*P(n,x) = (20*n-30+(48*n-72)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((20*n - 30 + (48*n - 72)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((20*n - 30 + (48*n - 72)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.