Real-rooted atlas

A038328 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 + 140\,x + 64\,x^{2} + 84\,n x + 64\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(686 + 3528\,x + 3808\,x^{2} + 1152\,x^{3} + 1176\,n x + 2016\,n x^{2} + 768\,n x^{3} + 224\,n^{2} x^{2} + 128\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 896\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01968965761668
127441881624192819253944
2384163512326773764587521024001628176
3537824614656015805440160563207168000117964846893792
4752953610326220833191424044957696030105600099090432128450561305274432
5105413504168661606465055191041101463552098344960004855431168125881548813421772835395144576
614757890562698585702412143635660824672783564827536588800018126943027270493667328150323855361358954496940146163968
720661046784425027248128218585441894451812845486086939220377600570998705356829607340277769470402887681712282664961342177280024554459049472

All 15 rows

Technical details
Node id
n-d04d98855e40597e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-18b5ed15f2b69ef1

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:           196,             896,             576;
  1:          2744,           18816,           24192,             8192;
  2:         38416,          351232,          677376,           458752,           102400;
  3:        537824,         6146560,        15805440,         16056320,          7168000,          1179648;
  4:       7529536,       103262208,       331914240,        449576960,        301056000,         99090432,         12845056;
  5:     105413504,      1686616064,      6505519104,      11014635520,       9834496000,       4855431168,       1258815488,        134217728;
  6:    1475789056,     26985857024,    121436356608,     246727835648,     275365888000,     181269430272,      70493667328,      15032385536,      1358954496;
  7:   20661046784,    425027248128,   2185854418944,    5181284548608,    6939220377600,    5709987053568,    2960734027776,     947040288768,    171228266496,     13421772800;
  8:  289254654976,   6611534970880,  38252452331520,  103625690972160,  161915142144000,  159879637499904,  103625690972160,   44195213475840,  11985978654720,   1879048192000,   129922760704;
  9: 4049565169664, 101817638551552, 654541962117120, 1994794551214080, 3562133127168000, 4103577362497536, 3191671281942528, 1701515718819840, 615280237608960, 144686710784000, 20008105148416, 1236950581248;
%F Formula
The row polynomials satisfy (49+(84*n+140)*x+(16*n^2+64*n+64)*x^2)*P(n,x) = (686+(1176*n+3528)*x+(224*n^2+2016*n+3808)*x^2+(128*n^2+768*n+1152)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 896*x + 576*x^2.
%t Mathematica
P[0, x_] := 196 + 896*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((686 + (1176*n + 3528)*x + (224*n^2 + 2016*n + 3808)*x^2 + (128*n^2 + 768*n + 1152)*x^3)*P[n - 1, x])/(49 + (84*n + 140)*x + (16*n^2 + 64*n + 64)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 196 + 896*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((686 + (1176*n + 3528)*x + (224*n^2 + 2016*n + 3808)*x^2 + (128*n^2 + 768*n + 1152)*x^3)*v[i - 1])/(49 + (84*n + 140)*x + (16*n^2 + 64*n + 64)*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.