Real-rooted atlas

A038334 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 4\,x + 2\,n x\right)\,P_{n}(x) = \left(100 + 160\,x + 48\,x^{2} + 40\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 640\,x + 192\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04006401921232
180001920011520204840768
2160000512000460800163840204801317120
3320000012800000153600008192000204800019660841796608
4640000003072000004608000003276800001228800002359296018350081307987968
5128000000071680000001290240000011468800000573440000016515072002569011201677721640478785536
625600000000163840000000344064000000367001600000229376000000880803840002055208960026843545601509949441241349423104
7512000000000368640000000088473600000001101004800000082575360000003963617280000123312537600024159191040027179089920134217728037780199833600

All 15 rows

Technical details
Node id
n-1fd1b3bc25cf1bde · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-65a6331fbe5d08da

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,              640,              192;
  1:            8000,            19200,            11520,             2048;
  2:          160000,           512000,           460800,           163840,            20480;
  3:         3200000,         12800000,         15360000,          8192000,          2048000,           196608;
  4:        64000000,        307200000,        460800000,        327680000,        122880000,         23592960,          1835008;
  5:      1280000000,       7168000000,      12902400000,      11468800000,       5734400000,       1651507200,        256901120,        16777216;
  6:     25600000000,     163840000000,     344064000000,     367001600000,     229376000000,      88080384000,      20552089600,      2684354560,       150994944;
  7:    512000000000,    3686400000000,    8847360000000,   11010048000000,    8257536000000,    3963617280000,    1233125376000,    241591910400,     27179089920,     1342177280;
  8:  10240000000000,   81920000000000,  221184000000000,  314572800000000,  275251200000000,  158544691200000,   61656268800000,  16106127360000,   2717908992000,   268435456000,   11811160064;
  9: 204800000000000, 1802240000000000, 5406720000000000, 8650752000000000, 8650752000000000, 5813305344000000, 2712875827200000, 885837004800000, 199313326080000, 29527900160000, 2598455214080, 103079215104;
%F Formula
The row polynomials satisfy (5+(2*n+4)*x)*P(n,x) = (100+(40*n+160)*x+(16*n+48)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 640*x + 192*x^2.
%t Mathematica
P[0, x_] := 400 + 640*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((100 + (40*n + 160)*x + (16*n + 48)*x^2)*P[n - 1, x])/(5 + (2*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 640*x + 192*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((100 + (40*n + 160)*x + (16*n + 48)*x^2)*v[i - 1])/(5 + (2*n + 4)*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.