Real-rooted atlas

A123519 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(9 + 6\,x + 6\,x^{2} + 21\,n + 16\,n x + 4\,n x^{2} + 16\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(30 + 39\,x + 40\,x^{2} + 14\,x^{3} + 62\,n + 101\,n x + 60\,n x^{2} + 12\,n x^{3} + 40\,n^{2} + 68\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(-15\,x - 42\,x^{2} - 28\,x^{3} - 8\,x^{4} - 16\,n x - 40\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 10 + 8\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0108119
1284415188
272176120241393
31765926442603511708
44161792273617924904817275
59605056100329360420084063130512
621761356833280408322640087361344801126417
748643507210272015724813499264944166322040991518612
8107528806430028855296059696038438414414429568297012012110211
92355221606484147218147842371840193356897697629515249764418014318527496
1051200520192227942456422408682240861696055444482269696566280800805720168134258649
111105921232896600576016794624297920003497702427535104144268804900896102960012412476441951136937340
122375682883584154624004823449697022464131833856123287808793804803466694499573761789216186368100102241544952795
13507904666828839038976134451200302643200467884032508234496390204672210234240779334401921462429914562722721288025512160291936
141081344152698889692774436539596891054080015798681601958584320175236300811296834565184537601655413763547315248372483884161632028818534425249
Technical details
Node id
n-24b2700579d99cff · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-36aebafdc60acded

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    10,      8,      1;
  1:    28,     44,     15,       1;
  2:    72,    176,    120,      24,       1;
  3:   176,    592,    644,     260,      35,       1;
  4:   416,   1792,   2736,    1792,     490,      48,      1;
  5:   960,   5056,  10032,    9360,    4200,     840,     63,      1;
  6:  2176,  13568,  33280,   40832,   26400,    8736,   1344,     80,     1;
  7:  4864,  35072, 102720,  157248,  134992,   64944,  16632,   2040,    99,    1;
  8: 10752,  88064, 300288,  552960,  596960,  384384, 144144,  29568,  2970,  120,   1;
  9: 23552, 216064, 841472, 1814784, 2371840, 1933568, 976976, 295152, 49764, 4180, 143, 1;
%F Formula
The row polynomials satisfy (4*n^3+16*n^2+21*n+9+(8*n^2+16*n+6)*x+(4*n+6)*x^2)*P(n,x) = (8*n^3+40*n^2+62*n+30+(12*n^3+68*n^2+101*n+39)*x+(24*n^2+60*n+40)*x^2+(12*n+14)*x^3)*P(n-1,x) - ((4*n^2+16*n+15)*x+(8*n^2+40*n+42)*x^2+(16*n+28)*x^3+8*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 10 + 8*x + x^2.
%t Mathematica
P[0, x_] := 10 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 + 40*n^2 + 62*n + 30 + (12*n^3 + 68*n^2 + 101*n + 39)*x + (24*n^2 + 60*n + 40)*x^2 + (12*n + 14)*x^3)*P[n - 1, x] - ((4*n^2 + 16*n + 15)*x + (8*n^2 + 40*n + 42)*x^2 + (16*n + 28)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(4*n^3 + 16*n^2 + 21*n + 9 + (8*n^2 + 16*n + 6)*x + (4*n + 6)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 10 + 8*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^3 + 40*n^2 + 62*n + 30 + (12*n^3 + 68*n^2 + 101*n + 39)*x + (24*n^2 + 60*n + 40)*x^2 + (12*n + 14)*x^3)*v[i - 1] - ((4*n^2 + 16*n + 15)*x + (8*n^2 + 40*n + 42)*x^2 + (16*n + 28)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(4*n^3 + 16*n^2 + 21*n + 9 + (8*n^2 + 16*n + 6)*x + (4*n + 6)*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

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023