Real-rooted atlas

A249253: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

A249253 ↗ nonnegative coefficients

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

Recurrence from A249253

\(P_{n}(x) = \left(2 + n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1314
212517
36028189
436018011551
52520132010513946
6201601092010201932119
7181440100800105002761293017

All 15 rows b-files: A249253 ?

Technical details
Node id
n-1d424c7e19773e5b · revision 0 · JSON
Definition source
oeis_catalog:A249253
Fingerprint
f6-063a87ccfc1d7ab7

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 ?

  • A249253 triangle 42 terms, 11 rows · tabf · nonn · catalog b-file?

Row sums ? A058279

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        3,        1;
  2:       12,        5;
  3:       60,       28,       1;
  4:      360,      180,      11;
  5:     2520,     1320,     105,     1;
  6:    20160,    10920,    1020,    19;
  7:   181440,   100800,   10500,   276,   1;
  8:  1814400,  1028160,  115920,  3780,  29;
  9: 19958400, 11491200, 1375920, 52080, 595, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+2)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(n + 2)*P[n - 1, x] + x*P[n - 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 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + 2)*v[i - 1] + x*v[i - 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

A249253 triangle, offset 0, from P0
%e Example
Triangle begins:
  0:        1;
  1:        3,        1;
  2:       12,        5;
  3:       60,       28,       1;
  4:      360,      180,      11;
  5:     2520,     1320,     105,     1;
  6:    20160,    10920,    1020,    19;
  7:   181440,   100800,   10500,   276,   1;
  8:  1814400,  1028160,  115920,  3780,  29;
  9: 19958400, 11491200, 1375920, 52080, 595, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+2)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(n + 2)*P[n - 1, x] + x*P[n - 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 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + 2)*v[i - 1] + x*v[i - 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

  • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 10

2 more

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