Real-rooted atlas

A008517 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(2 - 11\,x + 23\,x^{2} - 20\,x^{3} + 4\,x^{4} - 4\,n + 18\,n x - 38\,n x^{2} + 30\,n x^{3} - 4\,n x^{4} - 8\,n^{2} x + 28\,n^{2} x^{2} - 12\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(x - 6\,x^{2} + 12\,x^{3} - 9\,x^{4} + 2\,x^{5} - 2\,n x + 10\,n x^{2} - 14\,n x^{3} + 6\,n x^{4} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-7 + 15\,x - 11\,x^{2} + 2\,x^{3} + 6\,n - 18\,n x + 10\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 5\,x^{2} - 2\,x^{3} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2920635
32313813024315
45370816609241203465
511530181465219232730872045045
624111460101970264680228060642245040675675
7495403946073502771880457054028321206233764032011486475
810051355243254760241245006796582078597840371637366636960362880218243025
9202743972216197328184386160821551920158878680813742132885168992327699824036288004583103525
10407313941487642675012827227848556291240256645548323649784275224702013680762268953696752448039916800105411381075
1181674349418346799754832080748079796286696351797353488764304429952838778754816459294506640119065492224130967366404790016002635284526875
121635713415748152847752451186530380684286918896426064369852813415040542512222171124296961949758242984088623740298241966626380160190060335360622702080071152682225625
1332739410487146589997412302254038720549839861430846893669408768205734202612176488254897782848640049749723440461624155697952177749522252928342828146995202944310342400871782912002063427784543125
14655051248812842793998865017285965522804197204764618047865675331584028389504995399529288345368372960172862884587847201846141608340512011181680271686880370792251173606462946117235584048503818137600130767436800063966261320836875
Technical details
Node id
n-45531665af5787a9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7af5d5c3f437902e

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:    3,      2;
  2:    9,     20,        6;
  3:   23,    138,      130,        24;
  4:   53,    708,     1660,       924,       120;
  5:  115,   3018,    14652,     19232,      7308,        720;
  6:  241,  11460,   101970,    264680,    228060,      64224,       5040;
  7:  495,  40394,   607350,   2771880,   4570540,    2832120,     623376,     40320;
  8: 1005, 135524,  3254760,  24124500,  67965820,   78597840,   37163736,   6636960,   362880;
  9: 2027, 439722, 16197328, 184386160, 821551920, 1588786808, 1374213288, 516899232, 76998240, 3628800;
%F Formula
The row polynomials satisfy (2*n-3+2*x)*P(n,x) = -(4*n-2+(8*n^2-18*n+11)*x+(8*n^3-28*n^2+38*n-23)*x^2+(12*n^2-30*n+20)*x^3+(4*n-4)*x^4)*P(n-2,x) - ((2*n-1)*x+(4*n^2-10*n+6)*x^2-(4*n^2-14*n+12)*x^3-(6*n-9)*x^4-2*x^5)*d/dx P(n-2,x) + (6*n-7+(8*n^2-18*n+15)*x+(10*n-11)*x^2+2*x^3)*P(n-1,x) + ((2*n-3)*x-(2*n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 2 + (8*n^2 - 18*n + 11)*x + (8*n^3 - 28*n^2 + 38*n - 23)*x^2 + (12*n^2 - 30*n + 20)*x^3 + (4*n - 4)*x^4)*P[n - 2, x] - ((2*n - 1)*x + (4*n^2 - 10*n + 6)*x^2 - (4*n^2 - 14*n + 12)*x^3 - (6*n - 9)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (6*n - 7 + (8*n^2 - 18*n + 15)*x + (10*n - 11)*x^2 + 2*x^3)*P[n - 1, x] + ((2*n - 3)*x - (2*n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(2*n - 3 + 2*x)]];
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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 2 + (8*n^2 - 18*n + 11)*x + (8*n^3 - 28*n^2 + 38*n - 23)*x^2 + (12*n^2 - 30*n + 20)*x^3 + (4*n - 4)*x^4)*v[i - 2] - ((2*n - 1)*x + (4*n^2 - 10*n + 6)*x^2 - (4*n^2 - 14*n + 12)*x^3 - (6*n - 9)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (6*n - 7 + (8*n^2 - 18*n + 15)*x + (10*n - 11)*x^2 + 2*x^3)*v[i - 1] + ((2*n - 3)*x - (2*n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(2*n - 3 + 2*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

Same fingerprint ?

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.