Real-rooted atlas

basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2} + 3\,n x + 2\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 9\,x^{2} + 4\,x^{3} + 3\,n x + 7\,n x^{2} + 4\,n x^{3} + n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1 + 8\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
118918
2112271656
3116546425160
41209016012536432
5124135320375216491120
6128189560875756343642816
7132252896175020161372512816912
81363241344315045364116230472910016640

All 15 rows

Technical details
Node id
n-c1c8f7a2d47c2a75 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,  8,   9;
   2: 1, 12,  27,   16;
   3: 1, 16,  54,   64,   25;
   4: 1, 20,  90,  160,  125,    36;
   5: 1, 24, 135,  320,  375,   216,    49;
   6: 1, 28, 189,  560,  875,   756,   343,    64;
   7: 1, 32, 252,  896, 1750,  2016,  1372,   512,    81;
   8: 1, 36, 324, 1344, 3150,  4536,  4116,  2304,   729,  100;
   9: 1, 40, 405, 1920, 5250,  9072, 10290,  7680,  3645, 1000,  121;
  10: 1, 44, 495, 2640, 8250, 16632, 22638, 21120, 13365, 5500, 1331, 144;
%F Formula
The row polynomials satisfy (1+(3*n+2)*x+(n^2+2*n+1)*x^2)*P(n,x) = (1+(3*n+6)*x+(n^2+7*n+9)*x^2+(n^2+4*n+4)*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 8*x + 9*x^2.
%t Mathematica
P[1, x_] := 1 + 8*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (3*n + 6)*x + (n^2 + 7*n + 9)*x^2 + (n^2 + 4*n + 4)*x^3)*P[n - 1, x])/(1 + (3*n + 2)*x + (n^2 + 2*n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = ((1 + (3*n + 6)*x + (n^2 + 7*n + 9)*x^2 + (n^2 + 4*n + 4)*x^3)*v[i - 1])/(1 + (3*n + 2)*x + (n^2 + 2*n + 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.