Real-rooted atlas

A136124 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(2 + x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2127120
36047121120
4360342119181840
52520275411752452516720
6201602455212154313544533160480
7181440241128133938403697140742421604800

Show all 15 stored rows

Generated b-files: A001715 (row sums), A143492, A196845 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-634bc9ca21d13c4b · revision 1 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-a7859c04dfd745fc

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): yes

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:        3,        1;
  2:       12,        7,        1;
  3:       60,       47,       12,       1;
  4:      360,      342,      119,      18,       1;
  5:     2520,     2754,     1175,     245,      25,      1;
  6:    20160,    24552,    12154,    3135,     445,     33,     1;
  7:   181440,   241128,   133938,   40369,    7140,    742,    42,    1;
  8:  1814400,  2592720,  1580508,  537628,  111769,  14560,  1162,   52,  1;
  9: 19958400, 30334320, 19978308, 7494416, 1767087, 271929, 27342, 1734, 63, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+2+x)*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[(n + 2 + x)*P[n - 1, 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] = (n + 2 + x)*v[i - 1]); 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.

A143492: triangle, offset 3, from the row P0
%e Example
Triangle begins:
   3:        1;
   4:        3,        1;
   5:       12,        7,        1;
   6:       60,       47,       12,       1;
   7:      360,      342,      119,      18,       1;
   8:     2520,     2754,     1175,     245,      25,      1;
   9:    20160,    24552,    12154,    3135,     445,     33,     1;
  10:   181440,   241128,   133938,   40369,    7140,    742,    42,    1;
  11:  1814400,  2592720,  1580508,  537628,  111769,  14560,  1162,   52,  1;
  12: 19958400, 30334320, 19978308, 7494416, 1767087, 271929, 27342, 1734, 63, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n-1+x)*P(n-1,x) for n >= 4, with P(3,x) = 1.
%t Mathematica
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n - 1 + x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n - 2], {n, 3, 12}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 2)); v[1] = 1; for(n = 4, N, my(i = n - 2); v[i] = (n - 1 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};
T(12)

Generated with polytool c48629374597.

A196845: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  7,   12;
  3: 1, 12,   47,    60;
  4: 1, 18,  119,   342,    360;
  5: 1, 25,  245,  1175,   2754,    2520;
  6: 1, 33,  445,  3135,  12154,   24552,   20160;
  7: 1, 42,  742,  7140,  40369,  133938,  241128,   181440;
  8: 1, 52, 1162, 14560, 111769,  537628, 1580508,  2592720,  1814400;
  9: 1, 63, 1734, 27342, 271929, 1767087, 7494416, 19978308, 30334320, 19958400;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+2+x)*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[(n + 2 + x)*P[n - 1, 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] = (n + 2 + x)*v[i - 1]); 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.

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-3fcdea1a6c83c49b
  • 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-3fcdea1a6c83c49b
  • 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-3fcdea1a6c83c49b

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