Real-rooted atlas

A124860 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 5\,x + 5\,x^{2} - x^{3} + 6\,n x - 17\,n x^{2} + 5\,n x^{3} + 11\,n^{2} x^{2} - 11\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} - 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2} - 4\,x^{3} - 6\,x^{4} - 2\,x^{5} + 12\,n x + 34\,n x^{2} + 34\,n x^{3} + 14\,n x^{4} + 2\,n x^{5} + 22\,n^{2} x^{2} + 58\,n^{2} x^{3} + 52\,n^{2} x^{4} + 18\,n^{2} x^{5} + 2\,n^{2} x^{6} + 12\,n^{3} x^{3} + 28\,n^{3} x^{4} + 20\,n^{3} x^{5} + 4\,n^{3} x^{6} + 2\,n^{4} x^{4} + 4\,n^{4} x^{5} + 2\,n^{4} x^{6}\right)\,P_{n-2}(x) + \left(1 - x - 8\,x^{2} - 8\,x^{3} - x^{4} + x^{5} + 6\,n x - 12\,n x^{3} - 6\,n x^{4} + 11\,n^{2} x^{2} + 9\,n^{2} x^{3} - 4\,n^{2} x^{4} - 2\,n^{2} x^{5} + 6\,n^{3} x^{3} + 6\,n^{3} x^{4} + n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
23242754
356013580280
4111765947042751760
5214201890336026257569072
6431032580513760161259288210748160
785238016065476007437564260291555440239360
817154724309215321629925034473623461287552138511181952
934112276110484458304107415015467761403556785664248589341005674240
10683273202766151311360358575061961767028070524544024895356830008264326926592
111365600606756753603600112612502270268030900870288288001824322575075001816815196560125798400
12273113108816222149613120337961257786627212364875613842892810949944560082000218097664719168461539581681152
13546128397238336222498953697615375253019052459182724599748864569292867390461500188983366613379521199781710703562661822464
14109236116888945937636155522733480757872424561607286681239921510426569432892186784600132300468357253996816798481729972712245767512079964160
Technical details
Node id
n-1866672e0b3dba7d · 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:   3,    24,     27;
  3:   5,    60,    135,     80;
  4:  11,   176,    594,    704,     275;
  5:  21,   420,   1890,   3360,    2625,     756;
  6:  43,  1032,   5805,  13760,   16125,    9288,    2107;
  7:  85,  2380,  16065,  47600,   74375,   64260,   29155,   5440;
  8: 171,  5472,  43092, 153216,  299250,  344736,  234612,  87552,  13851;
  9: 341, 12276, 110484, 458304, 1074150, 1546776, 1403556, 785664, 248589, 34100;
%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) = (2+(12*n+6)*x+(22*n^2+34*n+4)*x^2+(12*n^3+58*n^2+34*n-4)*x^3+(2*n^4+28*n^3+52*n^2+14*n-6)*x^4+(4*n^4+20*n^3+18*n^2+2*n-2)*x^5+(2*n^4+4*n^3+2*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[((2 + (12*n + 6)*x + (22*n^2 + 34*n + 4)*x^2 + (12*n^3 + 58*n^2 + 34*n - 4)*x^3 + (2*n^4 + 28*n^3 + 52*n^2 + 14*n - 6)*x^4 + (4*n^4 + 20*n^3 + 18*n^2 + 2*n - 2)*x^5 + (2*n^4 + 4*n^3 + 2*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] = ((2 + (12*n + 6)*x + (22*n^2 + 34*n + 4)*x^2 + (12*n^3 + 58*n^2 + 34*n - 4)*x^3 + (2*n^4 + 28*n^3 + 52*n^2 + 14*n - 6)*x^4 + (4*n^4 + 20*n^3 + 18*n^2 + 2*n - 2)*x^5 + (2*n^4 + 4*n^3 + 2*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.