Real-rooted atlas

basis-map coefficients of A127647 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 5\,x + 5\,x^{2} - x^{3} + 6\,n x - 17\,n x^{2} + 5\,n x^{3} + 11\,n^{2} x^{2} - 11\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} - 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(1 + 3\,x + 2\,x^{2} - 2\,x^{3} - 3\,x^{4} - x^{5} + 6\,n x + 17\,n x^{2} + 17\,n x^{3} + 7\,n x^{4} + n x^{5} + 11\,n^{2} x^{2} + 29\,n^{2} x^{3} + 26\,n^{2} x^{4} + 9\,n^{2} x^{5} + n^{2} x^{6} + 6\,n^{3} x^{3} + 14\,n^{3} x^{4} + 10\,n^{3} x^{5} + 2\,n^{3} x^{6} + n^{4} x^{4} + 2\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-2}(x) + \left(1 - x - 8\,x^{2} - 8\,x^{3} - x^{4} + x^{5} + 6\,n x - 12\,n x^{3} - 6\,n x^{4} + 11\,n^{2} x^{2} + 9\,n^{2} x^{3} - 4\,n^{2} x^{4} - 2\,n^{2} x^{5} + 6\,n^{3} x^{3} + 6\,n^{3} x^{4} + n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
22161836
33368148168
4580270320125800
58160720128010002883456
613312175541604875280863714560
72158839691176018375158767203134459136
8341088856830464595006854446648174082754235008
95519801782073920173250249480226380126720400955500915200
108935603604517088046725080740891581068352032440589000107693508736
11144633671280380160118800023950083259872304128019245607920001916642073613271040
12233111841384028201602883375664329610549308118103049342135512600018607384026243937749627136
133771960426465417251526738875174671643169966841403648393011192695550013046462423446482826973892183758848
14610341604995903552640152652504396392089759670133985280148378230122122000738838103197376093811901673840137250674611200
Technical details
Node id
n-afcabc33edb2f88e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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): 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:  1;
  1:  1,    4;
  2:  2,   16,    18;
  3:  3,   36,    81,    48;
  4:  5,   80,   270,   320,    125;
  5:  8,  160,   720,  1280,   1000,    288;
  6: 13,  312,  1755,  4160,   4875,   2808,    637;
  7: 21,  588,  3969, 11760,  18375,  15876,   7203,   1344;
  8: 34, 1088,  8568, 30464,  59500,  68544,  46648,  17408,  2754;
  9: 55, 1980, 17820, 73920, 173250, 249480, 226380, 126720, 40095, 5500;
%F Formula
The row polynomials satisfy (1+(6*n-5)*x+(11*n^2-17*n+5)*x^2+(6*n^3-11*n^2+5*n-1)*x^3+(n^4-2*n^3+n^2)*x^4)*P(n,x) = (1+(6*n+3)*x+(11*n^2+17*n+2)*x^2+(6*n^3+29*n^2+17*n-2)*x^3+(n^4+14*n^3+26*n^2+7*n-3)*x^4+(2*n^4+10*n^3+9*n^2+n-1)*x^5+(n^4+2*n^3+n^2)*x^6)*P(n-2,x) + (1+(6*n-1)*x+(11*n^2-8)*x^2+(6*n^3+9*n^2-12*n-8)*x^3+(n^4+6*n^3-4*n^2-6*n-1)*x^4+(n^4-2*n^2+1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (6*n + 3)*x + (11*n^2 + 17*n + 2)*x^2 + (6*n^3 + 29*n^2 + 17*n - 2)*x^3 + (n^4 + 14*n^3 + 26*n^2 + 7*n - 3)*x^4 + (2*n^4 + 10*n^3 + 9*n^2 + n - 1)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*P[n - 2, x] + (1 + (6*n - 1)*x + (11*n^2 - 8)*x^2 + (6*n^3 + 9*n^2 - 12*n - 8)*x^3 + (n^4 + 6*n^3 - 4*n^2 - 6*n - 1)*x^4 + (n^4 - 2*n^2 + 1)*x^5)*P[n - 1, x])/(1 + (6*n - 5)*x + (11*n^2 - 17*n + 5)*x^2 + (6*n^3 - 11*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((1 + (6*n + 3)*x + (11*n^2 + 17*n + 2)*x^2 + (6*n^3 + 29*n^2 + 17*n - 2)*x^3 + (n^4 + 14*n^3 + 26*n^2 + 7*n - 3)*x^4 + (2*n^4 + 10*n^3 + 9*n^2 + n - 1)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*v[i - 2] + (1 + (6*n - 1)*x + (11*n^2 - 8)*x^2 + (6*n^3 + 9*n^2 - 12*n - 8)*x^3 + (n^4 + 6*n^3 - 4*n^2 - 6*n - 1)*x^4 + (n^4 - 2*n^2 + 1)*x^5)*v[i - 1])/(1 + (6*n - 5)*x + (11*n^2 - 17*n + 5)*x^2 + (6*n^3 - 11*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.