Real-rooted atlas

A136572 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14610
2329672200
33841728259212966000
4614436864829448294431104240000
512288092160027648004147200311040093312012000000
62949120265420809953280019906560022394880013436928033592320720000000
7825753608670412803901685760975421440014631321600131681894406584094720141087744050400000000
82642411520317089382401664719257604994157772809364045824001123685498880842764124160361184624640677221171204032000000000
995126814720128421199872077052719923202696845197312060679016939520910185254092809101852540928058511909191680219419659468803656994324480362880000000000
10380507258880057076088832000385263599616000154105439846400040452677959680007281482032742400910185254092800078015878922240004388393189376000146279772979200021941965946880036288000000000000
1116742319390720027624826994688002071862024601600093233791107072000279701373321216000587372883974553600881059325961830400943992134959104000707994101219328000353997050609664000106199115182899200144816975249408003991680000000000000
12803631330754560014465363953582080011933925261705216005966962630852608000201384988791275520004833239730990612480084581695292335718400108747893947288780800101951150575583232000679674337170554880003058534517267496960083414577743659008001042682221795737600479001600000000000000
13417888291992371200814882169385123840073339395244661145600403366673845636300800151262502692113612800040840875726870675456008168175145374135091200122522627180612026368001378379555781885296640011486496298182377472000689189777890942648320028194127277356744704007048531819339186176008132921330006753280062270208000000000000000
142340174435157278720049143663138302853120047915071559845281792002874904293590716907520011858980211061707243520035576940633185121730560080048116424666523893760013722534244228546953216001801082619554996787609600180108261955499678760960013508119646662475907072007368065261815895949312002763024473180960980992006376210322725294571520068316539172056727552008717829120000000000000000
Technical details
Node id
n-ae4b5c81684d82cb · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f06e51c139cfae05

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           4,             6;
  2:          32,            96,            72;
  3:         384,          1728,          2592,           1296;
  4:        6144,         36864,         82944,          82944,          31104;
  5:      122880,        921600,       2764800,        4147200,        3110400,         933120;
  6:     2949120,      26542080,      99532800,      199065600,      223948800,      134369280,       33592320;
  7:    82575360,     867041280,    3901685760,     9754214400,    14631321600,    13168189440,     6584094720,     1410877440;
  8:  2642411520,   31708938240,  166471925760,   499415777280,   936404582400,  1123685498880,   842764124160,   361184624640,    67722117120;
  9: 95126814720, 1284211998720, 7705271992320, 26968451973120, 60679016939520, 91018525409280, 91018525409280, 58511909191680, 21941965946880, 3656994324480;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+6*n*x)*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[(4*n + 6*n*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n + 6*n*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.