Real-rooted atlas

A269951: Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

A269951 ↗ nonnegative coefficients

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

Recurrence from A269951

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30213
4055111
5016239149
606511665141261
703266694701452011631
80195744293634141528027111743

All 15 rows b-files: A001339 (row sums), A269951 ?

Technical details
Node id
n-cd3918419c5686a8 · revision 0 · JSON
Definition source
oeis_catalog:A269951
Fingerprint
f6-56018d9771f15b64

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 -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 ?

  • A269951 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A001339

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0,      1;
   3: 0,      2,      1;
   4: 0,      5,      5,      1;
   5: 0,     16,     23,      9,      1;
   6: 0,     65,    116,     65,     14,     1;
   7: 0,    326,    669,    470,    145,    20,    1;
   8: 0,   1957,   4429,   3634,   1415,   280,   27,   1;
   9: 0,  13700,  33375,  30681,  14084,  3535,  490,  35,  1;
  10: 0, 109601, 283072, 284066, 147532, 43939, 7756, 798, 44, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n-1+x)*P(n-1,x) - (n-3)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := x;
P[n_, x_] := P[n, x] = Expand[(n - 1 + x)*P[n - 1, x] - (n - 3)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = (n - 1 + x)*v[i - 1] - (n - 3)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A269951 triangle, offset 0, from P1
%e Example
Triangle begins:
  0: 1;
  1: 0,      1;
  2: 0,      2,      1;
  3: 0,      5,      5,      1;
  4: 0,     16,     23,      9,      1;
  5: 0,     65,    116,     65,     14,     1;
  6: 0,    326,    669,    470,    145,    20,    1;
  7: 0,   1957,   4429,   3634,   1415,   280,   27,   1;
  8: 0,  13700,  33375,  30681,  14084,  3535,  490,  35,  1;
  9: 0, 109601, 283072, 284066, 147532, 43939, 7756, 798, 44, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[(n + x)*P[n - 1, x] - (n - 2)*P[n - 2, 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; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (n + x)*v[i - 1] - (n - 2)*v[i - 2]); 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)

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 12

4 more

Same fingerprint ?

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