Real-rooted atlas

A286785 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(42 + 84\,x + 26\,n + 94\,n x + 4\,n^{2} + 34\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(116 + 360\,x + 196\,x^{2} + 80\,n + 390\,n x + 238\,n x^{2} + 14\,n^{2} + 135\,n^{2} x + 94\,n^{2} x^{2} + 15\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2 + 22\,x + 46\,x^{2} - 70\,x^{3} + n + 10\,n x + 37\,n x^{2} - 48\,n x^{3} + n^{2} x + 7\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 14 + 28\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01428648
142144818275
2132660660176101638
3429286042902080325129996
4143012012245701820052505401462016
5486249504129948133280595001142483316389367
616796201552651168868224542640162792223441216182466750
75878681396031395605209344427329018098643910204032017012015737865

All 15 rows

Technical details
Node id
n-ed37a889ab35bb0d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9a1a7505fd4999af

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:     14,       28,        6;
  1:     42,      144,       81,         8;
  2:    132,      660,      660,       176,        10;
  3:    429,     2860,     4290,      2080,       325,        12;
  4:   1430,    12012,    24570,     18200,      5250,       540,       14;
  5:   4862,    49504,   129948,    133280,     59500,     11424,      833,       16;
  6:  16796,   201552,   651168,    868224,    542640,    162792,    22344,     1216,      18;
  7:  58786,   813960,  3139560,   5209344,   4273290,   1809864,   391020,    40320,    1701,     20;
  8: 208012,  3268760, 14709420,  29418840,  30284100,  16959096,  5206740,   850080,   68310,   2300,   22;
  9: 742900, 13075040, 67418175, 158631000, 198288750, 140263200, 57274140, 13358400, 1707750, 110000, 3025, 24;
%F Formula
The row polynomials satisfy (4*n^2+26*n+42+(4*n^3+34*n^2+94*n+84)*x)*P(n,x) = (14*n^2+80*n+116+(15*n^3+135*n^2+390*n+360)*x+(12*n^3+94*n^2+238*n+196)*x^2)*P(n-1,x) + (n+2+(n^2+10*n+22)*x+(7*n^2+37*n+46)*x^2-(8*n^2+48*n+70)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 28*x + 6*x^2.
%t Mathematica
P[0, x_] := 14 + 28*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^2 + 80*n + 116 + (15*n^3 + 135*n^2 + 390*n + 360)*x + (12*n^3 + 94*n^2 + 238*n + 196)*x^2)*P[n - 1, x] + (n + 2 + (n^2 + 10*n + 22)*x + (7*n^2 + 37*n + 46)*x^2 - (8*n^2 + 48*n + 70)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 26*n + 42 + (4*n^3 + 34*n^2 + 94*n + 84)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 14 + 28*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^2 + 80*n + 116 + (15*n^3 + 135*n^2 + 390*n + 360)*x + (12*n^3 + 94*n^2 + 238*n + 196)*x^2)*v[i - 1] + (n + 2 + (n^2 + 10*n + 22)*x + (7*n^2 + 37*n + 46)*x^2 - (8*n^2 + 48*n + 70)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 26*n + 42 + (4*n^3 + 34*n^2 + 94*n + 84)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.