Real-rooted atlas

A086645 → ⊙ C(n+1,k) (shift +2) → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(96 + 144\,x + 72\,x^{2} + 12\,x^{3} - 160\,n - 240\,n x - 120\,n x^{2} - 20\,n x^{3} + 64\,n^{2} + 96\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-3}(x) + \left(48 + 96\,x + 36\,x^{2} - 104\,n - 152\,n x - 14\,n x^{2} + 72\,n^{2} + 72\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(96\,x + 144\,x^{2} + 48\,x^{3}\right)\,P_{n-2}'(x) + \left(2 + x + 26\,n + 37\,n x + 6\,n^{2} - 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-48\,x - 48\,x^{2} + 54\,n x + 54\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 14 + 14\,x + x^{2}\)
\(P_{2}(x) = 92 + 138\,x + 48\,x^{2} + x^{3}\)
\(P_{3}(x) = 646 + 1292\,x + 762\,x^{2} + 116\,x^{3} + x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11414129
292138481279
3646129276211612817
446521163097603010230129283
5341241023721113905216094204021309869
625352888734811867527485102198002486464413321447
719016387606552120707729589384394121077442457708968135942657
814368844646597981187470081138688046042380417260362236913612121213861391820355

All 15 rows

Technical details
Node id
n-6173bcfb68499f9c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d122a18ed94aa5be

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 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        14,         14,           1;
   2:        92,        138,          48,           1;
   3:       646,       1292,         762,         116,           1;
   4:      4652,      11630,        9760,        3010,         230,          1;
   5:     34124,     102372,      111390,       52160,        9420,        402,          1;
   6:    253528,     887348,     1186752,      748510,      219800,      24864,        644,         1;
   7:   1901638,    7606552,    12070772,     9589384,     3941210,     774424,      57708,       968,        1;
   8:  14368844,   64659798,   118747008,   113868804,    60423804,   17260362,    2369136,    121212,     1386,      1;
   9: 109208164,  546040820,  1139132790,  1280286240,   834345120,  315404964,   65276610,   6463920,   235170,   1910,    1;
  10: 833981128, 4586896204, 10714557920, 13813789110, 10686883680, 5048017128, 1416460584, 218802210, 16046580, 427790, 2552, 1;
%F Formula
The row polynomials satisfy (2*n^2+3*n+1)*P(n,x) = (64*n^2-160*n+96+(96*n^2-240*n+144)*x+(48*n^2-120*n+72)*x^2+(8*n^2-20*n+12)*x^3)*P(n-3,x) + (72*n^2-104*n+48+(72*n^2-152*n+96)*x-(36*n^2+14*n-36)*x^2)*P(n-2,x) + (96*x+144*x^2+48*x^3)*d/dx P(n-2,x) + (6*n^2+26*n+2-(24*n^2-37*n-1)*x)*P(n-1,x) + ((54*n-48)*x+(54*n-48)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 14 + 14*x + x^2, P(2,x) = 92 + 138*x + 48*x^2 + x^3, P(3,x) = 646 + 1292*x + 762*x^2 + 116*x^3 + x^4.
%t Mathematica
P[1, x_] := 14 + 14*x + x^2;
P[2, x_] := 92 + 138*x + 48*x^2 + x^3;
P[3, x_] := 646 + 1292*x + 762*x^2 + 116*x^3 + x^4;
P[n_, x_] := P[n, x] = Expand[Together[((64*n^2 - 160*n + 96 + (96*n^2 - 240*n + 144)*x + (48*n^2 - 120*n + 72)*x^2 + (8*n^2 - 20*n + 12)*x^3)*P[n - 3, x] + (72*n^2 - 104*n + 48 + (72*n^2 - 152*n + 96)*x - (36*n^2 + 14*n - 36)*x^2)*P[n - 2, x] + (96*x + 144*x^2 + 48*x^3)*D[P[n - 2, x], x] + (6*n^2 + 26*n + 2 - (24*n^2 - 37*n - 1)*x)*P[n - 1, x] + ((54*n - 48)*x + (54*n - 48)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 3*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 14 + 14*x + x^2; v[2] = 92 + 138*x + 48*x^2 + x^3; v[3] = 646 + 1292*x + 762*x^2 + 116*x^3 + x^4; for(n = 4, N, my(i = n); v[i] = ((64*n^2 - 160*n + 96 + (96*n^2 - 240*n + 144)*x + (48*n^2 - 120*n + 72)*x^2 + (8*n^2 - 20*n + 12)*x^3)*v[i - 3] + (72*n^2 - 104*n + 48 + (72*n^2 - 152*n + 96)*x - (36*n^2 + 14*n - 36)*x^2)*v[i - 2] + (96*x + 144*x^2 + 48*x^3)*deriv(v[i - 2], x) + (6*n^2 + 26*n + 2 - (24*n^2 - 37*n - 1)*x)*v[i - 1] + ((54*n - 48)*x + (54*n - 48)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 3*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023