Real-rooted atlas

A116412 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(234 + 486\,x + 324\,x^{2} - 78\,n + 657\,n x + 810\,n x^{2} - 324\,n x^{3} - 243\,n x^{4} - 117\,n^{2} x + 837\,n^{2} x^{2} + 567\,n^{2} x^{3} - 243\,n^{2} x^{4} - 81\,n^{3} x^{2} + 243\,n^{3} x^{3}\right)\,P_{n}(x) = \left(1248 + 5400\,x + 6642\,x^{2} + 162\,x^{3} - 3402\,x^{4} - 1458\,x^{5} - 624\,n + 927\,n x + 9045\,n x^{2} + 10287\,n x^{3} + 1215\,n x^{4} - 1458\,n x^{5} - 1287\,n^{2} x - 1323\,n^{2} x^{2} + 3807\,n^{2} x^{3} + 3159\,n^{2} x^{4} - 648\,n^{3} x^{2} - 972\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(169 + 1092\,x + 1971\,x^{2} - 378\,x^{3} - 4131\,x^{4} - 3402\,x^{5} - 729\,x^{6} + 507\,n x + 3042\,n x^{2} + 5373\,n x^{3} + 1863\,n x^{4} - 1944\,n x^{5} - 729\,n x^{6} + 351\,n^{2} x^{2} + 1863\,n^{2} x^{3} + 2673\,n^{2} x^{4} + 729\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
213482788
343252324108727
41421140234917284055764
5469478213635159848100145844428
61549191526990311361691125349925103335440
7511674346331047690984767475463644142884174962492992

All 15 rows

Technical details
Node id
n-9f73b13f217c72f1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-69ea58d39e973037

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:     4,       6;
  2:    13,      48,      27;
  3:    43,     252,     324,      108;
  4:   142,    1140,    2349,     1728,      405;
  5:   469,    4782,   13635,    15984,     8100,     1458;
  6:  1549,   19152,   69903,   113616,    91125,    34992,     5103;
  7:  5116,   74346,  331047,   690984,   767475,   463644,   142884,   17496;
  8: 16897,  282180, 1483785,  3790368,  5410800,  4516884,  2178981,  559872,   59049;
  9: 55807, 1052964, 6386769, 19321416, 33804945, 36256086, 24111675, 9657792, 2125764, 196830;
%F Formula
The row polynomials satisfy -(78*n-234+(117*n^2-657*n-486)*x+(81*n^3-837*n^2-810*n-324)*x^2-(243*n^3+567*n^2-324*n)*x^3+(243*n^2+243*n)*x^4)*P(n,x) = -(624*n-1248+(1287*n^2-927*n-5400)*x+(648*n^3+1323*n^2-9045*n-6642)*x^2+(972*n^3-3807*n^2-10287*n-162)*x^3-(3159*n^2+1215*n-3402)*x^4+(1458*n+1458)*x^5)*P(n-1,x) + (169+(507*n+1092)*x+(351*n^2+3042*n+1971)*x^2+(1863*n^2+5373*n-378)*x^3+(2673*n^2+1863*n-4131)*x^4+(729*n^2-1944*n-3402)*x^5-(729*n+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[(-(624*n - 1248 + (1287*n^2 - 927*n - 5400)*x + (648*n^3 + 1323*n^2 - 9045*n - 6642)*x^2 + (972*n^3 - 3807*n^2 - 10287*n - 162)*x^3 - (3159*n^2 + 1215*n - 3402)*x^4 + (1458*n + 1458)*x^5)*P[n - 1, x] + (169 + (507*n + 1092)*x + (351*n^2 + 3042*n + 1971)*x^2 + (1863*n^2 + 5373*n - 378)*x^3 + (2673*n^2 + 1863*n - 4131)*x^4 + (729*n^2 - 1944*n - 3402)*x^5 - (729*n + 729)*x^6)*D[P[n - 1, x], x])/(-78*n + 234 - (117*n^2 - 657*n - 486)*x - (81*n^3 - 837*n^2 - 810*n - 324)*x^2 + (243*n^3 + 567*n^2 - 324*n)*x^3 - (243*n^2 + 243*n)*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] = (-(624*n - 1248 + (1287*n^2 - 927*n - 5400)*x + (648*n^3 + 1323*n^2 - 9045*n - 6642)*x^2 + (972*n^3 - 3807*n^2 - 10287*n - 162)*x^3 - (3159*n^2 + 1215*n - 3402)*x^4 + (1458*n + 1458)*x^5)*v[i - 1] + (169 + (507*n + 1092)*x + (351*n^2 + 3042*n + 1971)*x^2 + (1863*n^2 + 5373*n - 378)*x^3 + (2673*n^2 + 1863*n - 4131)*x^4 + (729*n^2 - 1944*n - 3402)*x^5 - (729*n + 729)*x^6)*deriv(v[i - 1], x))/(-78*n + 234 - (117*n^2 - 657*n - 486)*x - (81*n^3 - 837*n^2 - 810*n - 324)*x^2 + (243*n^3 + 567*n^2 - 324*n)*x^3 - (243*n^2 + 243*n)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.