Real-rooted atlas

A075499 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 5\,x - 3\,x^{2} - 4\,x^{3} - 4\,n x - 8\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 16 + 44\,x + 29\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
016442989
1643044402011009
2256198448164744165713457
310241305647744761765628415821210105
4409689088461056105766412383367271241703893747753
516384634880445440013757696226781442085737610133808203278574565473
665536470220843470848173705216378232576484557184366184288151535312265466731628999841
7262144358481924302110722169442304600023552010046430976105006822406724313824242072997237608565338704241897
8104857627865907243285872642711076864092772704256195047963648264432271104232946744832129245937712411330046045736591885993034281593
941943042192572416443773091843414974136321418720837632364373149286461527985070086990956086016531381906144025980914398247403637115929361461640927340167242321
1016777216173770014724643734159364355577610240216543724175366659059272908813619790623539219163622548684818708776953395212491383675270454547192086960140636822443121625661357673803154583649329
116710886413829459148849634806333445638657146880033175653711872012031753349365762920829679042560494881215034163259576808849018885092672120584192302917340511142411943368139120482810225548973562990532297942125050853217628313
122684354561103202615296541647888711687416791034757125119846450724864216443971229450246141776833236172812278120777200435217723259019635097618621596825850060814144681514146969675808135507108224272309013637193925890108765521636580513190237573826165199464341705
131073741824881193071411260252472161075299137315091251207977168335483699238955456783397683212768152869247385602965547047281033216502891069929308160063147885852764651525880613034940755968401642877827380377619577588424128975366455390261512855681291593450771459521185086954240540928707262835597618369
144294967296704363899125766816071491256320134627771883192320125677670479010201670387017984161873922639755544877019955270377615791160819712138040981892541382656202954234191703277568225288879906886885376188248981184026316800116733348789691077120521554375510199303681588786687708697952029562261109009808962536691399477677771047731789671001235265
Technical details
Node id
n-2e4481bb53b78f97 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-035ba7f125596294

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:      16,         44,          29;
  1:      64,        304,         440,          201;
  2:     256,       1984,        4816,         4744,          1657;
  3:    1024,      13056,       47744,        76176,         56284,         15821;
  4:    4096,      89088,      461056,      1057664,       1238336,        727124,        170389;
  5:   16384,     634880,     4454400,     13757696,      22678144,      20857376,      10133808,       2032785;
  6:   65536,    4702208,    43470848,    173705216,     378232576,     484557184,     366184288,     151535312,      26546673;
  7:  262144,   35848192,   430211072,   2169442304,    6000235520,   10046430976,   10500682240,    6724313824,    2420729972,     376085653;
  8: 1048576,  278659072,  4328587264,  27110768640,   92772704256,  195047963648,  264432271104,  232946744832,  129245937712,   41133004604,   5736591885;
  9: 4194304, 2192572416, 44377309184, 341497413632, 1418720837632, 3643731492864, 6152798507008, 6990956086016, 5313819061440, 2598091439824, 740363711592, 93614616409;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4-(4*n-5)*x-(8*n+3)*x^2-(4*n+4)*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 16 + 44*x + 29*x^2.
%t Mathematica
P[0, x_] := 16 + 44*x + 29*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4 - (4*n - 5)*x - (8*n + 3)*x^2 - (4*n + 4)*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 16 + 44*x + 29*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4 - (4*n - 5)*x - (8*n + 3)*x^2 - (4*n + 4)*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*deriv(v[i - 1], x))/(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

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.