Real-rooted atlas

A059364 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 4\,n x + 9\,n x^{2} + 4\,n^{2} + 12\,n^{2} x - 18\,n^{2} x^{2} - 4\,n^{3} - 8\,n^{3} x + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,n x^{2} + 4\,n x^{3} - 12\,n^{2} x^{2} - 8\,n^{2} x^{3} + 8\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 2\,n - 4\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
215441877
3105488423961112
494560728199387260019688
51039585692158499115472368504320411228
61351351363476321732931917921561600374544352809879156
7202702524235920697346828791475258334225213590884097772322560268026024
83445942547687544016217361542486332928209061622510351147683022388604830105632659208098940776
9654729075103011726604046971458673188104096746857650754676156686818422694248447056870461223169636288000269602835008
1013749310575242513788860108164039115622564117786562711965193075205746792130810233534615783345048880646943750077683204640004390848009799803782848
1131623414322561827868090803087994527087373049549669472101157276886375902792908483925434199654921122396486648960625399775636411343490776001208443262405748019200386118506004992
127905853580625169748909101800938933875616985248501090751542438986109784619754004980015806360283311166425401114116397856075524964965848901801210105455528001949185676275218694773043208095127040016388891405725184
13213458046676875499432493194110030316370232886245887950996282867681557392804773803751811164579435798921473599113256541678621935389832915236631293639609114112357484380396002429776014139848352035853000704307081956729601220496076800745435039121273600
14619028335362937515678907364019690010364959721098457553329598659461040592645883193821280392583920811747878052287724712154813298785519494249872065017625900066924346839689589701750173800002607970829504464165069191944368102466746823147912965339937289881601961511552000036167335916104761600
Technical details
Node id
n-e7607fd6407e2cb6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1489a2b47feb22bc

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:         3,           4;
  2:        15,          44,          18;
  3:       105,         488,         423,          96;
  4:       945,        6072,        8199,        3872,         600;
  5:     10395,       85692,      158499,      115472,       36850,        4320;
  6:    135135,     1363476,     3217329,     3191792,     1561600,      374544,       35280;
  7:   2027025,    24235920,    69734682,    87914752,    58334225,    21359088,     4097772,     322560;
  8:  34459425,   476875440,  1621736154,  2486332928,  2090616225,  1035114768,   302238860,   48301056,   3265920;
  9: 654729075, 10301172660, 40469714586, 73188104096, 74685765075, 46761566868, 18422694248, 4470568704, 612231696, 36288000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(8*n^3-12*n^2+4*n)*x-(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + ((8*n^3-12*n^2+4*n)*x^2+(4*n^3-8*n^2+4*n)*x^3)*d/dx P(n-2,x) + (4*n^2-2*n-1+(4*n^2-4*n)*x)*P(n-1,x) - (4*n^2-4*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + ((8*n^3 - 12*n^2 + 4*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*P[n - 1, x] - (4*n^2 - 4*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + ((8*n^3 - 12*n^2 + 4*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*v[i - 1] - (4*n^2 - 4*n)*x^2*deriv(v[i - 1], x))/(n - 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

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.