Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A008288 → D

nonnegative coefficients

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

Recurrence

\(\left(10 - 2\,x^{2} - 38\,n + 8\,n x + 6\,n x^{2} + 44\,n^{2} - 24\,n^{2} x - 4\,n^{2} x^{2} - 16\,n^{3} + 16\,n^{3} x\right)\,P_{n}(x) = \left(2 + 10\,x - 10\,x^{2} - 2\,x^{3} - 14\,n - 100\,n x - 110\,n x^{2} - 36\,n x^{3} - 4\,n x^{4} + 28\,n^{2} + 200\,n^{2} x + 172\,n^{2} x^{2} + 32\,n^{2} x^{3} - 16\,n^{3} - 80\,n^{3} x - 32\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} - 5\,x^{3} - x^{4} - 6\,n x - 44\,n x^{2} - 56\,n x^{3} - 20\,n x^{4} - 2\,n x^{5} + 8\,n^{2} x + 56\,n^{2} x^{2} + 56\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1325
272621458
31182189164556507
4151706931284115551010583940
5192901725515684157734402511601711028705
6234423477145963591553934502812919210431221025312200766
72762661413344411181524048634047332064820126783610225173756351141365175

All 15 rows

Technical details
Node id
n-f80541c1da09ba76 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2c8356228564163f

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  3,    2;
  2:  7,   26,    21,      4;
  3: 11,   82,   189,    164,     55,       6;
  4: 15,  170,   693,   1284,   1155,     510,     105,        8;
  5: 19,  290,  1725,   5156,   8415,    7734,    4025,     1160,      171,       10;
  6: 23,  442,  3477,  14596,  35915,   53934,   50281,    29192,    10431,     2210,     253,      12;
  7: 27,  626,  6141,  33444, 111815,  240486,  340473,   320648,   201267,    83610,   22517,    3756,     351,     14;
  8: 31,  842,  9909,  66564, 283475,  805470, 1569001,  2125832,  2017287,  1342450,  623645,  199692,   42939,   5894,   465,   16;
  9: 35, 1090, 14973, 119844, 622575, 2215830, 5568185, 10051720, 13163067, 12564650, 8750005, 4431660, 1618695, 419454, 74865, 8720, 595, 18;
%F Formula
The row polynomials satisfy -(16*n^3-44*n^2+38*n-10-(16*n^3-24*n^2+8*n)*x+(4*n^2-6*n+2)*x^2)*P(n,x) = -(16*n^3-28*n^2+14*n-2+(80*n^3-200*n^2+100*n-10)*x+(32*n^3-172*n^2+110*n+10)*x^2-(32*n^2-36*n-2)*x^3+4*n*x^4)*P(n-1,x) + ((8*n^2-6*n+1)*x+(56*n^2-44*n+5)*x^2+(56*n^2-56*n-5)*x^3+(8*n^2-20*n-1)*x^4-2*n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n^3 - 28*n^2 + 14*n - 2 + (80*n^3 - 200*n^2 + 100*n - 10)*x + (32*n^3 - 172*n^2 + 110*n + 10)*x^2 - (32*n^2 - 36*n - 2)*x^3 + 4*n*x^4)*P[n - 1, x] + ((8*n^2 - 6*n + 1)*x + (56*n^2 - 44*n + 5)*x^2 + (56*n^2 - 56*n - 5)*x^3 + (8*n^2 - 20*n - 1)*x^4 - 2*n*x^5)*D[P[n - 1, x], x])/(-16*n^3 + 44*n^2 - 38*n + 10 + (16*n^3 - 24*n^2 + 8*n)*x - (4*n^2 - 6*n + 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(16*n^3 - 28*n^2 + 14*n - 2 + (80*n^3 - 200*n^2 + 100*n - 10)*x + (32*n^3 - 172*n^2 + 110*n + 10)*x^2 - (32*n^2 - 36*n - 2)*x^3 + 4*n*x^4)*v[i - 1] + ((8*n^2 - 6*n + 1)*x + (56*n^2 - 44*n + 5)*x^2 + (56*n^2 - 56*n - 5)*x^3 + (8*n^2 - 20*n - 1)*x^4 - 2*n*x^5)*deriv(v[i - 1], x))/(-16*n^3 + 44*n^2 - 38*n + 10 + (16*n^3 - 24*n^2 + 8*n)*x - (4*n^2 - 6*n + 2)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.