Real-rooted atlas

A121574 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-33 - 18\,x + 27\,x^{2} + 42\,n - 9\,n x - 27\,n x^{2} - 9\,n^{2} + 27\,n^{2} x\right)\,P_{n}(x) = \left(-81 - 342\,x - 459\,x^{2} - 162\,x^{3} + 237\,n + 684\,n x + 351\,n x^{2} - 72\,n^{2} - 108\,n^{2} x\right)\,P_{n-1}(x) + \left(-26 - 177\,x - 405\,x^{2} - 351\,x^{3} - 81\,x^{4} + 39\,n + 207\,n x + 297\,n x^{2} + 81\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2211839
310519881384
4465136812153243372
519267722107736156121527792
67650388807443966744279454374220032
729523181818443799547236358425118098153091694208
811156780737223989773781404340564517496004745795248812781632
94149153450474120920852326255226807760189175507975989183708017714794935552
101523733143176865783553913153330818524376016727050896737571345371046908733590490696498432
11553906858035420265555422697572972116302351512841145469498469054644663121436662172539107019486175059160064
1219967220230800752118009078235178160686783462855887337633680261259755016225672212204391357868020091584999637729236446552064
137147253190347540451055905421703360599237321120980565115944866057944291746287907488250071334029313798770227013880532524189220726199260751249408
14254306505348986547021601105110797613593401958146585203371846616004186118526033790990414802502948935341189506675603954718201587113808721139940945669615661854527877120
Technical details
Node id
n-8724794021d32739 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e76770f74422271f

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 -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:     21,      18;
  3:    105,     198,       81;
  4:    465,    1368,     1215,      324;
  5:   1926,    7722,    10773,     6156,     1215;
  6:   7650,   38880,    74439,    66744,    27945,     4374;
  7:  29523,  181818,   443799,   547236,   358425,   118098,   15309;
  8: 111567,  807372,  2398977,  3781404,  3405645,  1749600,  474579,   52488;
  9: 414915, 3450474, 12092085, 23262552, 26807760, 18917550, 7975989, 1837080, 177147;
%F Formula
The row polynomials satisfy -(9*n^2-42*n+33-(27*n^2-9*n-18)*x+(27*n-27)*x^2)*P(n,x) = -(72*n^2-237*n+81+(108*n^2-684*n+342)*x-(351*n-459)*x^2+162*x^3)*P(n-1,x) + (39*n-26+(207*n-177)*x+(297*n-405)*x^2+(81*n-351)*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(72*n^2 - 237*n + 81 + (108*n^2 - 684*n + 342)*x - (351*n - 459)*x^2 + 162*x^3)*P[n - 1, x] + (39*n - 26 + (207*n - 177)*x + (297*n - 405)*x^2 + (81*n - 351)*x^3 - 81*x^4)*D[P[n - 1, x], x])/(-9*n^2 + 42*n - 33 + (27*n^2 - 9*n - 18)*x - (27*n - 27)*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; for(n = 2, N, my(i = n + 1); v[i] = (-(72*n^2 - 237*n + 81 + (108*n^2 - 684*n + 342)*x - (351*n - 459)*x^2 + 162*x^3)*v[i - 1] + (39*n - 26 + (207*n - 177)*x + (297*n - 405)*x^2 + (81*n - 351)*x^3 - 81*x^4)*deriv(v[i - 1], x))/(-9*n^2 + 42*n - 33 + (27*n^2 - 9*n - 18)*x - (27*n - 27)*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. [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.