Real-rooted atlas

A052553 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-100 + 40\,x - 44\,x^{2} - 4\,x^{3} + 100\,n - 165\,n x + 154\,n x^{2} + 17\,n x^{3} + 125\,n^{2} x - 185\,n^{2} x^{2} + 7\,n^{2} x^{3} - n^{2} x^{4} + 75\,n^{3} x^{2} - 35\,n^{3} x^{3} + 3\,n^{3} x^{4} + 15\,n^{4} x^{3} - 3\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-500 - 350\,x + 60\,x^{2} + 56\,x^{3} + 32\,x^{4} + 300\,n - 820\,n x - 888\,n x^{2} - 21\,n x^{3} + 16\,n x^{4} + 550\,n^{2} x - 140\,n^{2} x^{2} - 529\,n^{2} x^{3} - 73\,n^{2} x^{4} - 4\,n^{2} x^{5} + 300\,n^{3} x^{2} + 155\,n^{3} x^{3} - 61\,n^{3} x^{4} - 6\,n^{3} x^{5} + 55\,n^{4} x^{3} + 39\,n^{4} x^{4} + 3\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-125 - 150\,x + 65\,x^{2} + 134\,x^{3} + 60\,x^{4} + 16\,x^{5} - 375\,n x - 650\,n x^{2} - 240\,n x^{3} + 47\,n x^{4} + 12\,n x^{5} - 300\,n^{2} x^{2} - 590\,n^{2} x^{3} - 335\,n^{2} x^{4} - 49\,n^{2} x^{5} - 4\,n^{2} x^{6} - 75\,n^{3} x^{3} - 145\,n^{3} x^{4} - 78\,n^{3} x^{5} - 8\,n^{3} x^{6} - 5\,n^{4} x^{4} - 9\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3246
43811
5520934
68402775
713808116190
82115219864435
934284459224251026
10555209996401252339
118994020971680500365342
12144168042664096162521612027
1323329768478950447509724926962
143775232165242118412775352834359963
15610914031698457603237511340171564132702
Technical details
Node id
n-062c4929abd8e172 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9a99cf3d0e7eaf56

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
   2:  1;
   3:  2,   4;
   4:  3,   8;
   5:  5,  20,   9;
   6:  8,  40,  27;
   7: 13,  80,  81,  16;
   8: 21, 152, 198,  64;
   9: 34, 284, 459, 224,  25;
  10: 55, 520, 999, 640, 125;
%F Formula
The row polynomials satisfy (100*n-100+(125*n^2-165*n+40)*x+(75*n^3-185*n^2+154*n-44)*x^2+(15*n^4-35*n^3+7*n^2+17*n-4)*x^3+(n^5-3*n^4+3*n^3-n^2)*x^4)*P(n,x) = (300*n-500+(550*n^2-820*n-350)*x+(300*n^3-140*n^2-888*n+60)*x^2+(55*n^4+155*n^3-529*n^2-21*n+56)*x^3+(3*n^5+39*n^4-61*n^3-73*n^2+16*n+32)*x^4+(2*n^5-6*n^3-4*n^2)*x^5)*P(n-1,x) - (125+(375*n+150)*x+(300*n^2+650*n-65)*x^2+(75*n^3+590*n^2+240*n-134)*x^3+(5*n^4+145*n^3+335*n^2-47*n-60)*x^4+(9*n^4+78*n^3+49*n^2-12*n-16)*x^5+(4*n^4+8*n^3+4*n^2)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((300*n - 500 + (550*n^2 - 820*n - 350)*x + (300*n^3 - 140*n^2 - 888*n + 60)*x^2 + (55*n^4 + 155*n^3 - 529*n^2 - 21*n + 56)*x^3 + (3*n^5 + 39*n^4 - 61*n^3 - 73*n^2 + 16*n + 32)*x^4 + (2*n^5 - 6*n^3 - 4*n^2)*x^5)*P[n - 1, x] - (125 + (375*n + 150)*x + (300*n^2 + 650*n - 65)*x^2 + (75*n^3 + 590*n^2 + 240*n - 134)*x^3 + (5*n^4 + 145*n^3 + 335*n^2 - 47*n - 60)*x^4 + (9*n^4 + 78*n^3 + 49*n^2 - 12*n - 16)*x^5 + (4*n^4 + 8*n^3 + 4*n^2)*x^6)*D[P[n - 1, x], x])/(100*n - 100 + (125*n^2 - 165*n + 40)*x + (75*n^3 - 185*n^2 + 154*n - 44)*x^2 + (15*n^4 - 35*n^3 + 7*n^2 + 17*n - 4)*x^3 + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((300*n - 500 + (550*n^2 - 820*n - 350)*x + (300*n^3 - 140*n^2 - 888*n + 60)*x^2 + (55*n^4 + 155*n^3 - 529*n^2 - 21*n + 56)*x^3 + (3*n^5 + 39*n^4 - 61*n^3 - 73*n^2 + 16*n + 32)*x^4 + (2*n^5 - 6*n^3 - 4*n^2)*x^5)*v[i - 1] - (125 + (375*n + 150)*x + (300*n^2 + 650*n - 65)*x^2 + (75*n^3 + 590*n^2 + 240*n - 134)*x^3 + (5*n^4 + 145*n^3 + 335*n^2 - 47*n - 60)*x^4 + (9*n^4 + 78*n^3 + 49*n^2 - 12*n - 16)*x^5 + (4*n^4 + 8*n^3 + 4*n^2)*x^6)*deriv(v[i - 1], x))/(100*n - 100 + (125*n^2 - 165*n + 40)*x + (75*n^3 - 185*n^2 + 154*n - 44)*x^2 + (15*n^4 - 35*n^3 + 7*n^2 + 17*n - 4)*x^3 + (n^5 - 3*n^4 + 3*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, v[i])))};
T(10)

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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.