Real-rooted atlas

A104978 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 4\,n x - 2\,n^{2} - 10\,n^{2} x + 4\,n^{3} - 24\,n^{3} x - 8\,n^{3} x^{2} + 24\,n^{4} x - 8\,n^{4} x^{2} + 16\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-12\,x - 24\,x^{2} - 12\,x^{3} + 4\,n + 56\,n x + 102\,n x^{2} + 48\,n x^{3} - 12\,n^{2} - 40\,n^{2} x - 28\,n^{2} x^{2} + 8\,n^{3} - 86\,n^{3} x - 218\,n^{3} x^{2} - 120\,n^{3} x^{3} + 76\,n^{4} x + 88\,n^{4} x^{2} + 12\,n^{4} x^{3} + 56\,n^{5} x^{2} + 72\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 18\,x^{2} + 27\,x^{3} + 12\,x^{4} - 8\,n x - 44\,n x^{2} - 60\,n x^{3} - 24\,n x^{4} + n^{2} + n^{2} x - 41\,n^{2} x^{2} - 89\,n^{2} x^{3} - 48\,n^{2} x^{4} + 6\,n^{3} x + 52\,n^{3} x^{2} + 70\,n^{3} x^{3} + 24\,n^{3} x^{4} + 4\,n^{4} x^{2} + 40\,n^{4} x^{3} + 36\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
22202749
3584252192533
4143361620264013755985
542132089102288025025982868005
613251484504516016027300022276869972776225
7429200202162169900802320500293025618992404961288872869
814307779210024565643456169575002930256029248296156900483504303101427841
94862302328453492030387840111919500247118256336355404276883200126544275246675001158718085
10167961175720201455101569004806864396001853386920321516195035994816002517565050100150050017311651513224890641
11587864576264882565207845024003983801250127556629202700736038038457619200365046932252217608250078059810401211708160150780302645
1220801217829600382444920382444920022145248125822031610402061088029003576558585604311030438003548173200001903980592806008052960084613412521717198296289
13742900695354401642774770182673691201189281843755029535511001460428089120299749671936043972510467604589484900000333196603740016005453085444575633000125896279211219535560351013
142674440271455660700595122585801279200620930310000295066083312097535733094802316247464960040148813905200509432823900004684744248584430410360862336132184953336803453534966560410122467000222012772873345
Technical details
Node id
n-686dc44c06bd803d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-afe77babea4b595f

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:    1,      4;
  2:    2,     20,      27;
  3:    5,     84,     252,      192;
  4:   14,    336,    1620,     2640,      1375;
  5:   42,   1320,    8910,    22880,     25025,      9828;
  6:  132,   5148,   45045,   160160,    273000,    222768,     69972;
  7:  429,  20020,  216216,   990080,   2320500,   2930256,   1899240,    496128;
  8: 1430,  77792, 1002456,  5643456,  16957500,  29302560,  29248296,  15690048,   3504303;
  9: 4862, 302328, 4534920, 30387840, 111919500, 247118256, 336355404, 276883200, 126544275, 24667500;
%F Formula
The row polynomials satisfy (4*n^3-2*n^2-2*n+(24*n^4-24*n^3-10*n^2+4*n)*x+(16*n^5-8*n^4-8*n^3)*x^2)*P(n,x) = (8*n^3-12*n^2+4*n+(76*n^4-86*n^3-40*n^2+56*n-12)*x+(56*n^5+88*n^4-218*n^3-28*n^2+102*n-24)*x^2+(72*n^5+12*n^4-120*n^3+48*n-12)*x^3)*P(n-1,x) + (n^2+(6*n^3+n^2-8*n+3)*x+(4*n^4+52*n^3-41*n^2-44*n+18)*x^2+(40*n^4+70*n^3-89*n^2-60*n+27)*x^3+(36*n^4+24*n^3-48*n^2-24*n+12)*x^4)*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[Together[((8*n^3 - 12*n^2 + 4*n + (76*n^4 - 86*n^3 - 40*n^2 + 56*n - 12)*x + (56*n^5 + 88*n^4 - 218*n^3 - 28*n^2 + 102*n - 24)*x^2 + (72*n^5 + 12*n^4 - 120*n^3 + 48*n - 12)*x^3)*P[n - 1, x] + (n^2 + (6*n^3 + n^2 - 8*n + 3)*x + (4*n^4 + 52*n^3 - 41*n^2 - 44*n + 18)*x^2 + (40*n^4 + 70*n^3 - 89*n^2 - 60*n + 27)*x^3 + (36*n^4 + 24*n^3 - 48*n^2 - 24*n + 12)*x^4)*D[P[n - 1, x], x])/(4*n^3 - 2*n^2 - 2*n + (24*n^4 - 24*n^3 - 10*n^2 + 4*n)*x + (16*n^5 - 8*n^4 - 8*n^3)*x^2)]];
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] = ((8*n^3 - 12*n^2 + 4*n + (76*n^4 - 86*n^3 - 40*n^2 + 56*n - 12)*x + (56*n^5 + 88*n^4 - 218*n^3 - 28*n^2 + 102*n - 24)*x^2 + (72*n^5 + 12*n^4 - 120*n^3 + 48*n - 12)*x^3)*v[i - 1] + (n^2 + (6*n^3 + n^2 - 8*n + 3)*x + (4*n^4 + 52*n^3 - 41*n^2 - 44*n + 18)*x^2 + (40*n^4 + 70*n^3 - 89*n^2 - 60*n + 27)*x^3 + (36*n^4 + 24*n^3 - 48*n^2 - 24*n + 12)*x^4)*deriv(v[i - 1], x))/(4*n^3 - 2*n^2 - 2*n + (24*n^4 - 24*n^3 - 10*n^2 + 4*n)*x + (16*n^5 - 8*n^4 - 8*n^3)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.