Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A008288 → D+1

nonnegative coefficients

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

Recurrence

\(\left(12 - 7\,x - 10\,x^{2} + 4\,x^{3} + x^{4} - 20\,n - 10\,n x + 12\,n x^{2} + 2\,n x^{3} + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n}(x) = \left(9\,x^{3} - 2\,x^{4} - 6\,x^{5} - x^{6} + 4\,n x^{2} - 6\,n x^{3} - 12\,n x^{4} - 2\,n x^{5} - 8\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-2}(x) + \left(-27\,x - 78\,x^{2} - 25\,x^{3} + 23\,x^{4} + 10\,x^{5} + x^{6} - 12\,n - 50\,n x - 8\,n x^{2} + 48\,n x^{3} + 20\,n x^{4} + 2\,n x^{5} + 8\,n^{2} + 40\,n^{2} x + 40\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 5\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
145110
28333411187
3129323022796171676
416185778151514767411902314925
52030918705731970494175314173531629134446
62446536981575539564611175927036375140803369474351234227
7286536454354911201762628493805543692872413481059733087858036644111560200

All 15 rows

Technical details
Node id
n-a4933e776449d358 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-eedfb0e3e62b3e80

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,    5,     1;
  2:  8,   33,    34,     11,      1;
  3: 12,   93,   230,    227,     96,      17,       1;
  4: 16,  185,   778,   1515,   1476,     741,     190,       23,        1;
  5: 20,  309,  1870,   5731,   9704,    9417,    5314,     1735,      316,       29,        1;
  6: 24,  465,  3698,  15755,  39564,   61117,   59270,    36375,    14080,     3369,      474,      35,       1;
  7: 28,  653,  6454,  35491, 120176,  262849,  380554,   369287,   241348,   105973,    30878,    5803,     664,     41,      1;
  8: 32,  873, 10330,  69867, 300116,  862165, 1703246,  2349975,  2283016,  1566593,   757890,  256387,   59580,   9197,    886,    47,    1;
  9: 36, 1125, 15518, 124835, 652536, 2340345, 5937490, 10847175, 14419532, 14027213, 10006470, 5227115, 1988000, 543969, 104826, 13711, 1140, 53, 1;
%F Formula
The row polynomials satisfy (8*n^2-20*n+12+(8*n^2-10*n-7)*x+(12*n-10)*x^2+(2*n+4)*x^3+x^4)*P(n,x) = -((8*n^2-4*n)*x^2+(8*n^2+6*n-9)*x^3+(12*n+2)*x^4+(2*n+6)*x^5+x^6)*P(n-2,x) + (8*n^2-12*n+(40*n^2-50*n-27)*x+(40*n^2-8*n-78)*x^2+(8*n^2+48*n-25)*x^3+(20*n+23)*x^4+(2*n+10)*x^5+x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 5*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^2 - 4*n)*x^2 + (8*n^2 + 6*n - 9)*x^3 + (12*n + 2)*x^4 + (2*n + 6)*x^5 + x^6)*P[n - 2, x] + (8*n^2 - 12*n + (40*n^2 - 50*n - 27)*x + (40*n^2 - 8*n - 78)*x^2 + (8*n^2 + 48*n - 25)*x^3 + (20*n + 23)*x^4 + (2*n + 10)*x^5 + x^6)*P[n - 1, x])/(8*n^2 - 20*n + 12 + (8*n^2 - 10*n - 7)*x + (12*n - 10)*x^2 + (2*n + 4)*x^3 + 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; v[2] = 4 + 5*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^2 - 4*n)*x^2 + (8*n^2 + 6*n - 9)*x^3 + (12*n + 2)*x^4 + (2*n + 6)*x^5 + x^6)*v[i - 2] + (8*n^2 - 12*n + (40*n^2 - 50*n - 27)*x + (40*n^2 - 8*n - 78)*x^2 + (8*n^2 + 48*n - 25)*x^3 + (20*n + 23)*x^4 + (2*n + 10)*x^5 + x^6)*v[i - 1])/(8*n^2 - 20*n + 12 + (8*n^2 - 10*n - 7)*x + (12*n - 10)*x^2 + (2*n + 4)*x^3 + 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.