Real-rooted atlas

A081577 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(48 - 24\,x - 20\,x^{2} - 6\,x^{3} + 2\,x^{4} + 144\,n x - 12\,n x^{2} - 30\,n x^{3} - 4\,n x^{4} + 132\,n^{2} x^{2} + 30\,n^{2} x^{3} - 7\,n^{2} x^{4} - n^{2} x^{5} + 36\,n^{3} x^{3} + 6\,n^{3} x^{4} - n^{3} x^{5} + 3\,n^{4} x^{4}\right)\,P_{n}(x) = \left(96 + 336\,x + 440\,x^{2} + 260\,x^{3} + 60\,x^{4} - 4\,x^{5} - 4\,x^{6} + 288\,n x + 792\,n x^{2} + 780\,n x^{3} + 300\,n x^{4} + 14\,n x^{5} - 10\,n x^{6} + 264\,n^{2} x^{2} + 540\,n^{2} x^{3} + 334\,n^{2} x^{4} + 50\,n^{2} x^{5} - 8\,n^{2} x^{6} + 72\,n^{3} x^{3} + 108\,n^{3} x^{4} + 34\,n^{3} x^{5} - 2\,n^{3} x^{6} + 6\,n^{4} x^{4} + 6\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(96 + 144\,x + 32\,x^{2} - 80\,x^{3} - 42\,x^{4} - 5\,x^{5} - x^{6} + 288\,n x + 384\,n x^{2} + 168\,n x^{3} - 4\,n x^{4} - 12\,n x^{5} - 3\,n x^{6} + 264\,n^{2} x^{2} + 300\,n^{2} x^{3} + 100\,n^{2} x^{4} - 3\,n^{2} x^{6} + 72\,n^{3} x^{3} + 60\,n^{3} x^{4} + 10\,n^{3} x^{5} - n^{3} x^{6} + 6\,n^{4} x^{4} + 3\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
2624939
316969016218
444352522224251167
5120120024481728450365982
63283936102241011243507924929791
7896125443960050560308009216127464144954

All 15 rows

Technical details
Node id
n-dd48788568bd6ad3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3fc591d33e28ae54

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:    2,      4;
  2:    6,     24,      9;
  3:   16,     96,     90,     16;
  4:   44,    352,    522,    224,     25;
  5:  120,   1200,   2448,   1728,    450,     36;
  6:  328,   3936,  10224,  10112,   4350,    792,     49;
  7:  896,  12544,  39600,  50560,  30800,   9216,   1274,    64;
  8: 2448,  39168, 145584, 228096, 180900,  76896,  17346,  1920,   81;
  9: 6688, 120384, 514944, 956928, 937800, 521424, 166992, 29952, 2754, 100;
%F Formula
The row polynomials satisfy (48+(144*n-24)*x+(132*n^2-12*n-20)*x^2+(36*n^3+30*n^2-30*n-6)*x^3+(3*n^4+6*n^3-7*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (96+(288*n+336)*x+(264*n^2+792*n+440)*x^2+(72*n^3+540*n^2+780*n+260)*x^3+(6*n^4+108*n^3+334*n^2+300*n+60)*x^4+(6*n^4+34*n^3+50*n^2+14*n-4)*x^5-(2*n^3+8*n^2+10*n+4)*x^6)*P(n-2,x) + (96+(288*n+144)*x+(264*n^2+384*n+32)*x^2+(72*n^3+300*n^2+168*n-80)*x^3+(6*n^4+60*n^3+100*n^2-4*n-42)*x^4+(3*n^4+10*n^3-12*n-5)*x^5-(n^3+3*n^2+3*n+1)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((96 + (288*n + 336)*x + (264*n^2 + 792*n + 440)*x^2 + (72*n^3 + 540*n^2 + 780*n + 260)*x^3 + (6*n^4 + 108*n^3 + 334*n^2 + 300*n + 60)*x^4 + (6*n^4 + 34*n^3 + 50*n^2 + 14*n - 4)*x^5 - (2*n^3 + 8*n^2 + 10*n + 4)*x^6)*P[n - 2, x] + (96 + (288*n + 144)*x + (264*n^2 + 384*n + 32)*x^2 + (72*n^3 + 300*n^2 + 168*n - 80)*x^3 + (6*n^4 + 60*n^3 + 100*n^2 - 4*n - 42)*x^4 + (3*n^4 + 10*n^3 - 12*n - 5)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(48 + (144*n - 24)*x + (132*n^2 - 12*n - 20)*x^2 + (36*n^3 + 30*n^2 - 30*n - 6)*x^3 + (3*n^4 + 6*n^3 - 7*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)]];
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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((96 + (288*n + 336)*x + (264*n^2 + 792*n + 440)*x^2 + (72*n^3 + 540*n^2 + 780*n + 260)*x^3 + (6*n^4 + 108*n^3 + 334*n^2 + 300*n + 60)*x^4 + (6*n^4 + 34*n^3 + 50*n^2 + 14*n - 4)*x^5 - (2*n^3 + 8*n^2 + 10*n + 4)*x^6)*v[i - 2] + (96 + (288*n + 144)*x + (264*n^2 + 384*n + 32)*x^2 + (72*n^3 + 300*n^2 + 168*n - 80)*x^3 + (6*n^4 + 60*n^3 + 100*n^2 - 4*n - 42)*x^4 + (3*n^4 + 10*n^3 - 12*n - 5)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(48 + (144*n - 24)*x + (132*n^2 - 12*n - 20)*x^2 + (36*n^3 + 30*n^2 - 30*n - 6)*x^3 + (3*n^4 + 6*n^3 - 7*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)); 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

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