Real-rooted atlas

A196347 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 3\,n x\right)\,P_{n}(x) = \left(4\,n + 12\,n x + 9\,n x^{2} + 6\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1268
284854110
3484329726482100
438446081555220736972051000
53840576002592005184004860001749601500000
6460808294404665600124416001749600012597120367416051750000
76451201354752091445760304819200571536000617258880360067680881798402047500000
8103219202477260801950842880780337152018289152000263363788802304433152011287019520238085568091350000000
918579456050164531204514807808021069103104059256852480010666233446401244393902080914248581120385698620160714256704004536000000000
103715891200111476736000112870195200060197437440001975228416000042664933785600622196951040006094990540800038569862016000142851340800002357047123200248062500000000
1181749606400269773701120030349541376000182097248256000682864680960000172081899601920030114332430336003687469277184000311130220262400017285012236800005704054038144008485369643520014812875000000000
121961990553600706316599296008740667916288005827111944192000245831285145600007079941012193280014454879566561280021239823036579840022401375858892800016593611747328000082138378149273600244378645733376003309294160972800958938750000000000
13510117543936001989458421350400268576886882304001969563837136896009232330486579200002991275077651660800697964184785387520011965100310606643200151433300806115328001402160192649216000092542572714848256004129999112894054400111854142640880640013899035476085760066891825000000000000
1414283291230208005998982316687360087735116381552640070188093105242112003619073550739046400013028664782660567040034200245054483988480067004561739397201920098936423193328680960010992935910369853440009069172126055129088005396532174181564416002192341195761260544005448421906625617920062545659642385920005001684187500000000000
Technical details
Node id
n-7edc23b53fbb0465 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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:         2,          6;
  2:         8,         48,          54;
  3:        48,        432,         972,          648;
  4:       384,       4608,       15552,        20736,         9720;
  5:      3840,      57600,      259200,       518400,       486000,        174960;
  6:     46080,     829440,     4665600,     12441600,     17496000,      12597120,       3674160;
  7:    645120,   13547520,    91445760,    304819200,    571536000,     617258880,     360067680,     88179840;
  8:  10321920,  247726080,  1950842880,   7803371520,  18289152000,   26336378880,   23044331520,  11287019520,   2380855680;
  9: 185794560, 5016453120, 45148078080, 210691031040, 592568524800, 1066623344640, 1244393902080, 914248581120, 385698620160, 71425670400;
%F Formula
The row polynomials satisfy (2+3*n*x)*P(n,x) = (4*n+(6*n^2+12*n)*x+(9*n^2+9*n)*x^2)*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[Together[((4*n + (6*n^2 + 12*n)*x + (9*n^2 + 9*n)*x^2)*P[n - 1, x])/(2 + 3*n*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^2 + 12*n)*x + (9*n^2 + 9*n)*x^2)*v[i - 1])/(2 + 3*n*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.