Real-rooted atlas

A081577 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,x^{2} + 3\,n - 3\,n x - 2\,n x^{2} + 3\,n^{2} x\right)\,P_{n}(x) = \left(-24 - 66\,x - 44\,x^{2} - 8\,x^{3} + 39\,n + 87\,n x + 28\,n x^{2} - 9\,n^{2} - 15\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 42\,x - 63\,x^{2} - 32\,x^{3} - 4\,x^{4} + 9\,n + 36\,n x + 39\,n x^{2} + 6\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
212820
3488024152
4176464336641040
56002176259211521606680
6196890881516811136352038441264
76272352007584078848409609984896248000
8195841294083421444631043417601359362688020481460864
96019245772814353922400768231744013086724193286963246088473760
1018272015717125704704113802241363584010091520458035212247041751041024048557120
115491205271296217416965042995272360960666501123965337614974976342835243008022528275512448
12163660817344256801369602120232963549696003926016002910781441439744004641177692774401036288491521550539520
1348439045617049628741785685456076816368691202117763072188856729611644928004910100481377894402442035224576001064968666469248
1414251776179497984100768665633270128647180195840106530693121113287884882643025924342837248159154176039487078462816256575078422937648156942080
Technical details
Node id
n-94f7da70b8d9535f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ccfe222bc030f3af

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     2;
  2:    12,      8;
  3:    48,     80,      24;
  4:   176,    464,     336,      64;
  5:   600,   2176,    2592,    1152,     160;
  6:  1968,   9088,   15168,   11136,    3520,     384;
  7:  6272,  35200,   75840,   78848,   40960,    9984,    896;
  8: 19584, 129408,  342144,  463104,  341760,  135936,  26880,  2048;
  9: 60192, 457728, 1435392, 2400768, 2317440, 1308672, 419328, 69632, 4608;
%F Formula
The row polynomials satisfy (3*n-3+(3*n^2-3*n)*x-(2*n-2)*x^2)*P(n,x) = -(9*n^2-39*n+24+(15*n^2-87*n+66)*x-(28*n-44)*x^2+8*x^3)*P(n-1,x) + (9*n-9+(36*n-42)*x+(39*n-63)*x^2+(6*n-32)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 39*n + 24 + (15*n^2 - 87*n + 66)*x - (28*n - 44)*x^2 + 8*x^3)*P[n - 1, x] + (9*n - 9 + (36*n - 42)*x + (39*n - 63)*x^2 + (6*n - 32)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(3*n - 3 + (3*n^2 - 3*n)*x - (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 39*n + 24 + (15*n^2 - 87*n + 66)*x - (28*n - 44)*x^2 + 8*x^3)*v[i - 1] + (9*n - 9 + (36*n - 42)*x + (39*n - 63)*x^2 + (6*n - 32)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(3*n - 3 + (3*n^2 - 3*n)*x - (2*n - 2)*x^2)); 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. [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.