Real-rooted atlas

A056242 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(10 + 8\,x - 13\,n - 12\,n x + 2\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(2 - 16\,x - 16\,x^{2} - 5\,n + 14\,n x + 16\,n x^{2} + 10\,n^{2} + 10\,n^{2} x + n^{3}\right)\,P_{n-1}(x) + \left(-8 - 28\,x - 28\,x^{2} - 8\,x^{3} + 12\,n + 34\,n x + 30\,n x^{2} + 8\,n x^{3} + 4\,n^{2} + 6\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
222
313821
4658024169
5292538324641218
612363030274210881608256
75036154041860311104328038453811
8199807322811059588192392009216896341307
9777283319526017526006763568401262402464020482121876
102978561452304307183236835322740410129590438057663488460812990510
111127824618128014942808209219921870594511079912434952010915841589761024078570081
1242292002573609669992952112072616117043525833336164097816813746176301132838912022528470555325
1315731904105242976318022560573170608684791260569631966336732172141495040414282248051200934912491522795281974
145812499242396892814092291202823273424379791374036149074262496891054126132915246258963212014336020979200221184010649616491668364
152135177601686425920611491368013480666720201637927402161360923617084313239100846693124446310752144641664033750604853501952516505622937696731038431
Technical details
Node id
n-b7674b057a88e854 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f8e29e91a9c12eec

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:      2;
   3:     13,       8;
   4:     65,      80,      24;
   5:    292,     538,     324,      64;
   6:   1236,    3030,    2742,    1088,     160;
   7:   5036,   15404,   18603,   11104,    3280,     384;
   8:  19980,   73228,  110595,   88192,   39200,    9216,    896;
   9:  77728,  331952,  601752,  600676,  356840,  126240,  24640,  2048;
  10: 297856, 1452304, 3071832, 3683532, 2740410, 1295904, 380576, 63488, 4608;
%F Formula
The row polynomials satisfy (n^3+2*n^2-13*n+10+(4*n^2-12*n+8)*x)*P(n,x) = (n^3+10*n^2-5*n+2+(10*n^2+14*n-16)*x+(16*n-16)*x^2)*P(n-1,x) + (4*n^2+12*n-8+(6*n^2+34*n-28)*x+(2*n^2+30*n-28)*x^2+(8*n-8)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 10*n^2 - 5*n + 2 + (10*n^2 + 14*n - 16)*x + (16*n - 16)*x^2)*P[n - 1, x] + (4*n^2 + 12*n - 8 + (6*n^2 + 34*n - 28)*x + (2*n^2 + 30*n - 28)*x^2 + (8*n - 8)*x^3)*D[P[n - 1, x], x])/(n^3 + 2*n^2 - 13*n + 10 + (4*n^2 - 12*n + 8)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((n^3 + 10*n^2 - 5*n + 2 + (10*n^2 + 14*n - 16)*x + (16*n - 16)*x^2)*v[i - 1] + (4*n^2 + 12*n - 8 + (6*n^2 + 34*n - 28)*x + (2*n^2 + 30*n - 28)*x^2 + (8*n - 8)*x^3)*deriv(v[i - 1], x))/(n^3 + 2*n^2 - 13*n + 10 + (4*n^2 - 12*n + 8)*x)); 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

References

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