Real-rooted atlas

A145596 → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(\left(4\,n + n^{2}\right)\,P_{n}(x) = \left(6 + 2\,x + 10\,n + 5\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21414331
34872324156
4165330225605785
55721430132055010063978
620026006700740041155154720335
770722475234944254801019221842248104856

Show all 15 stored rows

Generated b-files: A259476 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-3c983d83f25bade9 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-db69b92786c145d5

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     4,      2;
  2:    14,     14,      3;
  3:    48,     72,     32,      4;
  4:   165,    330,    225,     60,      5;
  5:   572,   1430,   1320,    550,    100,      6;
  6:  2002,   6006,   7007,   4004,   1155,    154,     7;
  7:  7072,  24752,  34944,  25480,  10192,   2184,   224,    8;
  8: 25194, 100776, 167076, 148512,  76440,  22932,  3822,  312,   9;
  9: 90440, 406980, 775200, 813960, 514080, 199920, 47040, 6300, 420, 10;
%F Formula
The row polynomials satisfy (n^2+4*n)*P(n,x) = (4*n^2+10*n+6+(3*n^2+5*n+2)*x)*P(n-1,x) - ((2*n+2)*x+(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 10*n + 6 + (3*n^2 + 5*n + 2)*x)*P[n - 1, x] - ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (3*n^2 + 5*n + 2)*x)*v[i - 1] - ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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)

Generated with polytool c48629374597.

A259476: reversed triangle, offset 4, from the row P0
%e Example
Triangle begins:
   4:  1;
   5:  2,   4;
   6:  3,  14,   14;
   7:  4,  32,   72,    48;
   8:  5,  60,  225,   330,    165;
   9:  6, 100,  550,  1320,   1430,    572;
  10:  7, 154, 1155,  4004,   7007,   6006,   2002;
  11:  8, 224, 2184, 10192,  25480,  34944,  24752,   7072;
  12:  9, 312, 3822, 22932,  76440, 148512, 167076, 100776,  25194;
  13: 10, 420, 6300, 47040, 199920, 514080, 813960, 775200, 406980, 90440;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2-4*n)*P(n,x) = (4*n^2-22*n+30+(3*n^2-19*n+30)*x)*P(n-1,x) - ((2*n-6)*x+(2*n-6)*x^2)*d/dx P(n-1,x) for n >= 5, with P(4,x) = 1.
%t Mathematica
P[4, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 22*n + 30 + (3*n^2 - 19*n + 30)*x)*P[n - 1, x] - ((2*n - 6)*x + (2*n - 6)*x^2)*D[P[n - 1, x], x])/(n^2 - 4*n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 3]], {n, 4, 13}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 3)); v[1] = 1; for(n = 5, N, my(i = n - 3); v[i] = ((4*n^2 - 22*n + 30 + (3*n^2 - 19*n + 30)*x)*v[i - 1] - ((2*n - 6)*x + (2*n - 6)*x^2)*deriv(v[i - 1], x))/(n^2 - 4*n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 3 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 3, v[i])))};
T(13)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-33d7b30ac5eafba9
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-33d7b30ac5eafba9
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-33d7b30ac5eafba9
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-33d7b30ac5eafba9

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

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