Real-rooted atlas

A038334 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 12\,x + 6\,n x\right)\,P_{n}(x) = \left(100 + 480\,x + 432\,x^{2} + 120\,n x + 144\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1920\,x + 1728\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0400192017284048
180005760010368055296224576
2160000153600041472004423680165888011925760
332000003840000013824000022118400016588800047775744614687744
4640000009216000004147200000884736000099532800005733089280133772083231004250112
5128000000021504000000116121600000309657600000464486400000401316249600187280916480366917713921538338537472
62560000000049152000000030965760000009909043200000185794560000002140353331200014982473318400587068342272099067782758475349563080704
75120000000001105920000000079626240000000297271296000000668860416000000963158999040000898948399104000528361508044800178322008965120264180754022403652538142556160

All 15 rows

Technical details
Node id
n-1d55f39abb48a994 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-572def99d14f8fb6

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             400,             1920,              1728;
  1:            8000,            57600,            103680,              55296;
  2:          160000,          1536000,           4147200,            4423680,            1658880;
  3:         3200000,         38400000,         138240000,          221184000,          165888000,            47775744;
  4:        64000000,        921600000,        4147200000,         8847360000,         9953280000,          5733089280,          1337720832;
  5:      1280000000,      21504000000,      116121600000,       309657600000,       464486400000,        401316249600,        187280916480,         36691771392;
  6:     25600000000,     491520000000,     3096576000000,      9909043200000,     18579456000000,      21403533312000,      14982473318400,       5870683422720,        990677827584;
  7:    512000000000,   11059200000000,    79626240000000,    297271296000000,    668860416000000,     963158999040000,     898948399104000,     528361508044800,     178322008965120,     26418075402240;
  8:  10240000000000,  245760000000000,  1990656000000000,   8493465600000000,  22295347200000000,   38526359961600000,   44947419955200000,   35224100536320000,   17832200896512000,   5283615080448000,    697437190619136;
  9: 204800000000000, 5406720000000000, 48660480000000000, 233570304000000000, 700710912000000000, 1412633198592000000, 1977686478028800000, 1937325529497600000, 1307694732410880000, 581197658849280000, 153436181936209920, 18260173718028288;
%F Formula
The row polynomials satisfy (5+(6*n+12)*x)*P(n,x) = (100+(120*n+480)*x+(144*n+432)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1920*x + 1728*x^2.
%t Mathematica
P[0, x_] := 400 + 1920*x + 1728*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((100 + (120*n + 480)*x + (144*n + 432)*x^2)*P[n - 1, x])/(5 + (6*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1920*x + 1728*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((100 + (120*n + 480)*x + (144*n + 432)*x^2)*v[i - 1])/(5 + (6*n + 12)*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.