Real-rooted atlas

A334218 → D → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(6 - 5\,n + n^{2}\right)\,P_{n}(x) = \left(-16 - 32\,x - 16\,x^{2} + 36\,n + 72\,n x + 36\,n x^{2} - 28\,n^{2} - 56\,n^{2} x - 28\,n^{2} x^{2} + 9\,n^{3} + 18\,n^{3} x + 9\,n^{3} x^{2} - n^{4} - 2\,n^{4} x - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x - 8\,x^{2} - 4\,x^{3} + 8\,n x + 16\,n x^{2} + 8\,n x^{3} - 5\,n^{2} x - 10\,n^{2} x^{2} - 5\,n^{2} x^{3} + n^{3} x + 2\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-15 - 18\,x + 26\,n + 30\,n x - 13\,n^{2} - 14\,n^{2} x + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2} + 4\,n x + 4\,n x^{2} - n^{2} x - n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
10
20
322
49615
5485612116
6300500225201045
72160468032407443010854
817640470404410016856220542127883
9161280510720604800326592772806096561686824
1016329605987520857304060056642097900326700160657224639921
111814400075600000127008000109569600510300001227780013062004088090394976570
1221954240010245312001975881600202710816011846142003908203206733650050161761012771106894951943
1328740096001484904960032332608000384718118402710350720011436353280278087040035222457618689616245640132130219369884
1440475635200229361932800556539984000753746273280622312891200320978211840101788727040187412184961777836060680225005855851562645791318157
156102480384003763196236800100690926336001529688416256014493390912000885343979520034806739968008525848531201211869058408729065800243002760137608818257549670979150
Technical details
Node id
n-f232d2f66d74f533 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ef5562416e43169f

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
  • 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:       0;
   2:       0;
   3:       2;
   4:       9,       6;
   5:      48,      56,      12;
   6:     300,     500,     225,      20;
   7:    2160,    4680,    3240,     744,      30;
   8:   17640,   47040,   44100,   16856,    2205,     42;
   9:  161280,  510720,  604800,  326592,   77280,   6096,    56;
  10: 1632960, 5987520, 8573040, 6005664, 2097900, 326700, 16065, 72;
%F Formula
The row polynomials satisfy (n^2-5*n+6)*P(n,x) = -(n^4-9*n^3+28*n^2-36*n+16+(2*n^4-18*n^3+56*n^2-72*n+32)*x+(n^4-9*n^3+28*n^2-36*n+16)*x^2)*P(n-2,x) + ((n^3-5*n^2+8*n-4)*x+(2*n^3-10*n^2+16*n-8)*x^2+(n^3-5*n^2+8*n-4)*x^3)*d/dx P(n-2,x) + (2*n^3-13*n^2+26*n-15+(2*n^3-14*n^2+30*n-18)*x)*P(n-1,x) - ((n^2-4*n+3)*x+(n^2-4*n+3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 9*n^3 + 28*n^2 - 36*n + 16 + (2*n^4 - 18*n^3 + 56*n^2 - 72*n + 32)*x + (n^4 - 9*n^3 + 28*n^2 - 36*n + 16)*x^2)*P[n - 2, x] + ((n^3 - 5*n^2 + 8*n - 4)*x + (2*n^3 - 10*n^2 + 16*n - 8)*x^2 + (n^3 - 5*n^2 + 8*n - 4)*x^3)*D[P[n - 2, x], x] + (2*n^3 - 13*n^2 + 26*n - 15 + (2*n^3 - 14*n^2 + 30*n - 18)*x)*P[n - 1, x] - ((n^2 - 4*n + 3)*x + (n^2 - 4*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 - 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = (-(n^4 - 9*n^3 + 28*n^2 - 36*n + 16 + (2*n^4 - 18*n^3 + 56*n^2 - 72*n + 32)*x + (n^4 - 9*n^3 + 28*n^2 - 36*n + 16)*x^2)*v[i - 2] + ((n^3 - 5*n^2 + 8*n - 4)*x + (2*n^3 - 10*n^2 + 16*n - 8)*x^2 + (n^3 - 5*n^2 + 8*n - 4)*x^3)*deriv(v[i - 2], x) + (2*n^3 - 13*n^2 + 26*n - 15 + (2*n^3 - 14*n^2 + 30*n - 18)*x)*v[i - 1] - ((n^2 - 4*n + 3)*x + (n^2 - 4*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 - 5*n + 6)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.