Real-rooted atlas

A038334 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 50\,x + 16\,x^{2} + 30\,n x + 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(500 + 1800\,x + 1360\,x^{2} + 288\,x^{3} + 600\,n x + 720\,n x^{2} + 192\,n x^{3} + 80\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1280\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
040012805762256
180003840034560819289152
2160000102400013824006553601024003324160
33200000256000004608000032768000102400001179648119067648
46400000061440000013824000001310720000614400000141557760128450564140322816
512800000001433600000038707200000458752000002867200000099090432001798307840134217728140711968768
6256000000003276800000001032192000000146800640000011468800000005284823040001438646272002147483648013589544964695539122176
75120000000007372800000000265420800000004404019200000041287680000000237817036800008631877632000193273528320024461180928013421772800154359102177280

All 15 rows

Technical details
Node id
n-36a0dd106a540a34 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2e8ee919cea99b33

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:             400,             1280,               576;
  1:            8000,            38400,             34560,              8192;
  2:          160000,          1024000,           1382400,            655360,            102400;
  3:         3200000,         25600000,          46080000,          32768000,          10240000,           1179648;
  4:        64000000,        614400000,        1382400000,        1310720000,         614400000,         141557760,          12845056;
  5:      1280000000,      14336000000,       38707200000,       45875200000,       28672000000,        9909043200,        1798307840,        134217728;
  6:     25600000000,     327680000000,     1032192000000,     1468006400000,     1146880000000,      528482304000,      143864627200,      21474836480,       1358954496;
  7:    512000000000,    7372800000000,    26542080000000,    44040192000000,    41287680000000,    23781703680000,     8631877632000,    1932735283200,     244611809280,     13421772800;
  8:  10240000000000,  163840000000000,   663552000000000,  1258291200000000,  1376256000000000,   951268147200000,   431593881600000,  128849018880000,   24461180928000,   2684354560000,   129922760704;
  9: 204800000000000, 3604480000000000, 16220160000000000, 34603008000000000, 43253760000000000, 34879832064000000, 18990130790400000, 7086696038400000, 1793819934720000, 295279001600000, 28583007354880, 1236950581248;
%F Formula
The row polynomials satisfy (25+(30*n+50)*x+(4*n^2+16*n+16)*x^2)*P(n,x) = (500+(600*n+1800)*x+(80*n^2+720*n+1360)*x^2+(32*n^2+192*n+288)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1280*x + 576*x^2.
%t Mathematica
P[0, x_] := 400 + 1280*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((500 + (600*n + 1800)*x + (80*n^2 + 720*n + 1360)*x^2 + (32*n^2 + 192*n + 288)*x^3)*P[n - 1, x])/(25 + (30*n + 50)*x + (4*n^2 + 16*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1280*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((500 + (600*n + 1800)*x + (80*n^2 + 720*n + 1360)*x^2 + (32*n^2 + 192*n + 288)*x^3)*v[i - 1])/(25 + (30*n + 50)*x + (4*n^2 + 16*n + 16)*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. [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.