Real-rooted atlas

basis-map coefficients of A113214, shifted → x ↦ x+1 → (θ+1)²

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

Recurrence

\(\left(4 - 2\,x + 6\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-8 - 12\,x - 6\,x^{2} - x^{3} - 12\,n x - 10\,n x^{2} - 2\,n x^{3} - 2\,n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1-2-4-6
2416929
3-8-48-54-16-126
41612821612825513
5-32-320-720-640-250-36-1998
664768216025601500432497533
7-128-1792-6048-8960-7000-3024-686-64-27702
8256409616128286722800016128548810248199873
9-512-9216-41472-86016-100800-72576-32928-9216-1458-100-354294
10102420480103680245760336000290304164640614401458020001211240029
11-2048-45056-253440-675840-1056000-1064448-724416-337920-106920-22000-2662-144-4290894
124096983046082561802240316800036495362897664162201664152017600031944345616914703201
13-8192-212992-1437696-4685824-9152000-11860992-10762752-7028736-3335904-1144000-276848-44928-4394-196-49955454
1416384458752335462411927552256256003690086437669632281149441556755264064001937936419328615165488225168466797
Technical details
Node id
n-2fd8ff26d4d48fb5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • 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:    1;
  1:   -2,    -4;
  2:    4,    16,      9;
  3:   -8,   -48,    -54,    -16;
  4:   16,   128,    216,    128,      25;
  5:  -32,  -320,   -720,   -640,    -250,    -36;
  6:   64,   768,   2160,   2560,    1500,    432,     49;
  7: -128, -1792,  -6048,  -8960,   -7000,  -3024,   -686,   -64;
  8:  256,  4096,  16128,  28672,   28000,  16128,   5488,  1024,    81;
  9: -512, -9216, -41472, -86016, -100800, -72576, -32928, -9216, -1458, -100;
%F Formula
The row polynomials satisfy (4+(6*n-2)*x+n^2*x^2)*P(n,x) = -(8+(12*n+12)*x+(2*n^2+10*n+6)*x^2+(n^2+2*n+1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8 + (12*n + 12)*x + (2*n^2 + 10*n + 6)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(4 + (6*n - 2)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(8 + (12*n + 12)*x + (2*n^2 + 10*n + 6)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(4 + (6*n - 2)*x + n^2*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.