Real-rooted atlas

A038301 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(400 + 1100\,x + 484\,x^{2} + 660\,n x + 484\,n x^{2} + 121\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8000 + 39600\,x + 41140\,x^{2} + 11979\,x^{3} + 13200\,n x + 21780\,n x^{2} + 7986\,n x^{3} + 2420\,n^{2} x^{2} + 1331\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1760\,x + 1089\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0400176010893249
18000528006534021296147436
21600001408000261360017036803660256251305
33200000352000008712000085184000366025005797836253104336
464000000844800000261360000034073600002196150000695740320868064899908456809
51280000000197120000007318080000011925760000010248700000048701822400121529084601247178944378019309804
62560000000045056000000019514880000003816243200000409948000000025974305280009722326768001995486310401736306936114129946105201
7512000000000101376000000005018112000000011448729600000014758128000000011688437376000058333960608000179593767936003125352484980235794769100519438154415680

All 15 rows

Technical details
Node id
n-65720707fe82f0eb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8bc1b8fcc230ea6a

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,             1760,              1089;
  1:            8000,            52800,             65340,             21296;
  2:          160000,          1408000,           2613600,           1703680,             366025;
  3:         3200000,         35200000,          87120000,          85184000,           36602500,            5797836;
  4:        64000000,        844800000,        2613600000,        3407360000,         2196150000,          695740320,           86806489;
  5:      1280000000,      19712000000,       73180800000,      119257600000,       102487000000,        48701822400,        12152908460,        1247178944;
  6:     25600000000,     450560000000,     1951488000000,     3816243200000,      4099480000000,      2597430528000,       972232676800,      199548631040,       17363069361;
  7:    512000000000,   10137600000000,    50181120000000,   114487296000000,    147581280000000,    116884373760000,     58333960608000,    17959376793600,     3125352484980,     235794769100;
  8:  10240000000000,  225280000000000,  1254528000000000,  3271065600000000,   4919376000000000,   4675374950400000,   2916698030400000,  1197291786240000,   312535248498000,   47158953820000,   3138428376721;
  9: 204800000000000, 4956160000000000, 30666240000000000, 89954304000000000, 154608960000000000, 171430414848000000, 128334713337600000, 65851048243200000, 22919251556520000, 5187484920200000, 690454242878620, 41084880567984;
%F Formula
The row polynomials satisfy (400+(660*n+1100)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (8000+(13200*n+39600)*x+(2420*n^2+21780*n+41140)*x^2+(1331*n^2+7986*n+11979)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1760*x + 1089*x^2.
%t Mathematica
P[0, x_] := 400 + 1760*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8000 + (13200*n + 39600)*x + (2420*n^2 + 21780*n + 41140)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*P[n - 1, x])/(400 + (660*n + 1100)*x + (121*n^2 + 484*n + 484)*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 + 1760*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8000 + (13200*n + 39600)*x + (2420*n^2 + 21780*n + 41140)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*v[i - 1])/(400 + (660*n + 1100)*x + (121*n^2 + 484*n + 484)*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

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.