Real-rooted atlas

A113413 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(48 - 828\,x - 1188\,x^{2} - 81\,x^{3} + 243\,x^{4} - 432\,n - 252\,n x - 540\,n x^{2} + 243\,n x^{4} + 216\,n^{2} - 108\,n^{2} x + 162\,n^{2} x^{2} + 243\,n^{2} x^{3} + 162\,n^{3} x\right)\,P_{n}(x) = \left(-624 - 6246\,x - 14418\,x^{2} - 5589\,x^{3} + 2673\,x^{4} + 1458\,x^{5} - 1608\,n - 4086\,n x + 810\,n x^{2} + 2592\,n x^{3} + 729\,n x^{4} + 1440\,n^{2} + 4158\,n^{2} x + 1134\,n^{2} x^{2} - 729\,n^{2} x^{3} - 216\,n^{3} - 486\,n^{3} x\right)\,P_{n-1}(x) + \left(-80 - 1032\,x - 3906\,x^{2} - 4347\,x^{3} + 1620\,x^{4} + 2916\,x^{5} + 729\,x^{6} - 240\,n - 1296\,n x - 1350\,n x^{2} + 1944\,n x^{3} + 2673\,n x^{4} + 729\,n x^{5} + 144\,n^{2} + 864\,n^{2} x + 1458\,n^{2} x^{2} + 486\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
22536970
38623116227506
426910921404648813494
5798437184787074243024323394
62285158164155352056319958748729153182
7638253487178308302589277830134865306182187986266
81749717234469825615059521870290135010854091810497665616266902
947286535539255819667187881060678810335762613380620907723542941968339400914
101263371617852891008127645408531935106585202852642548264889447853517118098059049245570254
113344064778535298254061068870872432413803670898943724679702519467741099492382883559538972341771471519430666
12878309138562209669510039337984810352346751848805236228629709019607767201148089707442080180103926240127545845314419343305350
13229179039566499305353314139098432641602677308598062925125843727061314760539697493441945028120765173167097436882005441452398159432357149507394
145946581111525120943291161475833506415952925457374999271846354099958578563233584709256631514627268596821310725051663663520812919330711339231324782969347952532286
Technical details
Node id
n-3d46518983f7437c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-34729af0efddbc5d

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:     1;
  1:     6,      3;
  2:    25,     36,       9;
  3:    86,    231,     162,      27;
  4:   269,   1092,    1404,     648,       81;
  5:   798,   4371,    8478,    7074,     2430,      243;
  6:  2285,  15816,   41553,   52056,    31995,     8748,     729;
  7:  6382,  53487,  178308,  302589,   277830,   134865,   30618,    2187;
  8: 17497, 172344,  698256, 1505952,  1870290,  1350108,  540918,  104976,   6561;
  9: 47286, 535539, 2558196, 6718788, 10606788, 10335762, 6133806, 2090772, 354294, 19683;
%F Formula
The row polynomials satisfy (216*n^2-432*n+48+(162*n^3-108*n^2-252*n-828)*x+(162*n^2-540*n-1188)*x^2+(243*n^2-81)*x^3+(243*n+243)*x^4)*P(n,x) = -(216*n^3-1440*n^2+1608*n+624+(486*n^3-4158*n^2+4086*n+6246)*x-(1134*n^2+810*n-14418)*x^2+(729*n^2-2592*n+5589)*x^3-(729*n+2673)*x^4-1458*x^5)*P(n-1,x) + (144*n^2-240*n-80+(864*n^2-1296*n-1032)*x+(1458*n^2-1350*n-3906)*x^2+(486*n^2+1944*n-4347)*x^3+(2673*n+1620)*x^4+(729*n+2916)*x^5+729*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(216*n^3 - 1440*n^2 + 1608*n + 624 + (486*n^3 - 4158*n^2 + 4086*n + 6246)*x - (1134*n^2 + 810*n - 14418)*x^2 + (729*n^2 - 2592*n + 5589)*x^3 - (729*n + 2673)*x^4 - 1458*x^5)*P[n - 1, x] + (144*n^2 - 240*n - 80 + (864*n^2 - 1296*n - 1032)*x + (1458*n^2 - 1350*n - 3906)*x^2 + (486*n^2 + 1944*n - 4347)*x^3 + (2673*n + 1620)*x^4 + (729*n + 2916)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(216*n^2 - 432*n + 48 + (162*n^3 - 108*n^2 - 252*n - 828)*x + (162*n^2 - 540*n - 1188)*x^2 + (243*n^2 - 81)*x^3 + (243*n + 243)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(216*n^3 - 1440*n^2 + 1608*n + 624 + (486*n^3 - 4158*n^2 + 4086*n + 6246)*x - (1134*n^2 + 810*n - 14418)*x^2 + (729*n^2 - 2592*n + 5589)*x^3 - (729*n + 2673)*x^4 - 1458*x^5)*v[i - 1] + (144*n^2 - 240*n - 80 + (864*n^2 - 1296*n - 1032)*x + (1458*n^2 - 1350*n - 3906)*x^2 + (486*n^2 + 1944*n - 4347)*x^3 + (2673*n + 1620)*x^4 + (729*n + 2916)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(216*n^2 - 432*n + 48 + (162*n^3 - 108*n^2 - 252*n - 828)*x + (162*n^2 - 540*n - 1188)*x^2 + (243*n^2 - 81)*x^3 + (243*n + 243)*x^4)); 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.