Real-rooted atlas

A185411: A triangular decomposition of the double factorial numbers A001147.

A185410 → reverse

A185411 nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A185411
\(P_{n}(x) = \left(-x + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20213
30410115
40860361105
50162965161161945
6032132851683508358110395
7064566442960642402112010861135135

Show all 15 stored rows

Generated b-files: A001147 (row sums), A156919, A185410 (reversed triangle), A185411 ?

Computed in your browser ?

Technical details
Node id
n-84fc1826f57fd756 · revision 1 · JSON
Definition source
oeis_catalog:A185411
Fingerprint
f6-b79b1b87544662ee

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: 0,   1;
  2: 0,   2,     1;
  3: 0,   4,    10,       1;
  4: 0,   8,    60,      36,        1;
  5: 0,  16,   296,     516,      116,        1;
  6: 0,  32,  1328,    5168,     3508,      358,       1;
  7: 0,  64,  5664,   42960,    64240,    21120,    1086,      1;
  8: 0, 128, 23488,  320064,   900560,   660880,  118632,   3272,    1;
  9: 0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1)*x*P(n-1,x) + (2*x-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[(2*n - 1)*x*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
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] = (2*n - 1)*x*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], 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)

Generated with polytool c48629374597.

A156919: triangle, offset 0, from the row P1, low zeros dropped
%e Example
Triangle begins:
  0:   1;
  1:   2,      1;
  2:   4,     10,        1;
  3:   8,     60,       36,         1;
  4:  16,    296,      516,       116,         1;
  5:  32,   1328,     5168,      3508,       358,         1;
  6:  64,   5664,    42960,     64240,     21120,      1086,        1;
  7: 128,  23488,   320064,    900560,    660880,    118632,     3272,       1;
  8: 256,  95872,  2225728,  10725184,  14713840,   6049744,   638968,    9832,     1;
  9: 512, 387840, 14792448, 114735936, 265115424, 205021488, 51293760, 3352152, 29514, 1;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (2*n+1)*x*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = x.
%t Mathematica
P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[(2*n + 1)*x*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1)*x*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A185410: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,    0;
  2: 1,    2,      0;
  3: 1,   10,      4,       0;
  4: 1,   36,     60,       8,        0;
  5: 1,  116,    516,     296,       16,        0;
  6: 1,  358,   3508,    5168,     1328,       32,       0;
  7: 1, 1086,  21120,   64240,    42960,     5664,      64,     0;
  8: 1, 3272, 118632,  660880,   900560,   320064,   23488,   128,   0;
  9: 1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2*n-1)*x*P(n-1,x) + (2*x-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[(2*n - 1)*x*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (2*n - 1)*x*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A185411: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0,   1;
  2: 0,   2,     1;
  3: 0,   4,    10,       1;
  4: 0,   8,    60,      36,        1;
  5: 0,  16,   296,     516,      116,        1;
  6: 0,  32,  1328,    5168,     3508,      358,       1;
  7: 0,  64,  5664,   42960,    64240,    21120,    1086,      1;
  8: 0, 128, 23488,  320064,   900560,   660880,  118632,   3272,    1;
  9: 0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1)*x*P(n-1,x) + (2*x-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[(2*n - 1)*x*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (2*n - 1)*x*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-5c085bd25a6bcbdf
  • 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-5c085bd25a6bcbdf
  • 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-5c085bd25a6bcbdf

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 (1)

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