Real-rooted atlas

A085881 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 - 4\,x + 12\,n + 8\,n x\right)\,P_{n-1}(x) + \left(-4\,x + 8\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
210819248348
3324011760720096023160
41360809139201034880322560268802434320
57348320871516801632960008951040016934400967680365208480
648498912099100108802898225792024908083200803658240010218700804257792073386371520
737829151360131352084864057900915072007335568880640359368289280076329437184069742632960221405184018905944337280
834046236224001992978763776001294730467430400232800908820480016260494874624005133057785856007718184714240053137244160001328431104006047425736352000
93472716094848003411215339051520032169810684211200080041245277593600076644712630886400033745251928227840074267534986444800831969831813120044714990960640090333315072002343513046754880000
10395889634812672006507701786518732800881642200776809472002981901149693706240003812062452080762880002249515261249191936006797888640772669440011018857772482560000955139306913792000411919916728320006865331945472001079054159142037478400
11498820939863966720013699428481013603328002646692765290577203200012012165794494729543680020116525494355837747200015484485786168428789760061722626631361029734400135776909078015901696001687785219805741056000116563038436196352000412331836645048320057668788341964800581082476245155082598400
1268837289701227407360031554577032798772592640086483854462609837449216005216532373327551133777920011284081153237569018101760011110442845609457489018880056834339298914065003315200162962359399860582481920002724271845627963978547200267876135351493853184000150862703678346756096004456643963067039744005305528527460761600361213443771404010077491200
131032559345518411110400007895527987169815888896000030584894832241430833152000002433923422460007847649280000067278978716134180733337600000835019596496990426967244800005373619731979500624150528000019535234478879556011294720000421768184734517160443904000055369712668186494369792000044136824458159262269440000206570753216684752896000051728903142742425600000530552852746076160000256346786173611615389107200000
14167274613973982599884800002132907720629010174001152000011646361948848694866531532800001215918469734384559257354240000042592350653640511831063756800000658845053754312194923010457600005254453147120145166686552064000023736854562149293092451123200000643714689909513811537428480000010834021811705133794328576000001146172256586752176580198400007555482209906444489195520000298517154256137989652480000641756730681653723136000057299708096576225280000205746435539474382869965608960000
Technical details
Node id
n-392ebb75e65f97c1 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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 0)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1;
  1:               6,                 4;
  2:             108,               192,                 48;
  3:            3240,             11760,               7200,                960;
  4:          136080,            913920,            1034880,             322560,              26880;
  5:         7348320,          87151680,          163296000,           89510400,           16934400,             967680;
  6:       484989120,        9910010880,        28982257920,        24908083200,         8036582400,         1021870080,          42577920;
  7:     37829151360,     1313520848640,      5790091507200,      7335568880640,      3593682892800,       763294371840,       69742632960,       2214051840;
  8:   3404623622400,   199297876377600,   1294730467430400,   2328009088204800,   1626049487462400,    513305778585600,    77181847142400,    5313724416000,    132843110400;
  9: 347271609484800, 34112153390515200, 321698106842112000, 800412452775936000, 766447126308864000, 337452519282278400, 74267534986444800, 8319698318131200, 447149909606400, 9033331507200;
%F Formula
The row polynomials satisfy P(n,x) = (12*n-6+(8*n-4)*x)*P(n-1,x) + (8*n-4)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(12*n - 6 + (8*n - 4)*x)*P[n - 1, x] + (8*n - 4)*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (12*n - 6 + (8*n - 4)*x)*v[i - 1] + (8*n - 4)*x*deriv(v[i - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.