A008299: Triangle T(n,k) of associated Stirling numbers of second kind, n >= 2, 1 <= k <= floor(n/2).

A008299 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence from A008299

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

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	1	3			4
3	1	10			11
4	1	25	15		41
5	1	56	105		162
6	1	119	490	105	715
7	1	246	1918	1260	3425

All 15 rows b-files: A008299 ?

Technical details

Node id: n-61f790c1436a0d7c · revision 0 · JSON
Definition source: oeis_catalog:A008299
Fingerprint: f6-e8edac49e7142220

Observed on the stored rows ?

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

Matches ?

A008299 triangle 47 terms, 12 rows · tabf · nonn · catalog b-file?

Row sums ? A000296

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,    3;
  3: 1,   10;
  4: 1,   25,    15;
  5: 1,   56,   105;
  6: 1,  119,   490,   105;
  7: 1,  246,  1918,  1260;
  8: 1,  501,  6825,  9450,   945;
  9: 1, 1012, 22935, 56980, 17325;

%F Formula

The row polynomials satisfy P(n,x) = P(n-1,x) + x*d/dx P(n-1,x) + (n+1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + x*D[P[n - 1, x], x] + (n + 1)*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + x*deriv(v[i - 1], x) + (n + 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

A008299 triangle, offset 2, from P₀

%e Example

Triangle begins:
   2: 1;
   3: 1;
   4: 1,    3;
   5: 1,   10;
   6: 1,   25,    15;
   7: 1,   56,   105;
   8: 1,  119,   490,   105;
   9: 1,  246,  1918,  1260;
  10: 1,  501,  6825,  9450,   945;
  11: 1, 1012, 22935, 56980, 17325;

%F Formula

The row polynomials satisfy P(n,x) = P(n-1,x) + x*d/dx P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 4, with P(2,x) = 1, P(3,x) = 1.

%t Mathematica

P[2, x_] := 1;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + x*D[P[n - 1, x], x] + (n - 1)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 2, 11}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N - 1)); v[1] = 1; v[2] = 1; for(n = 4, N, my(i = n - 1); v[i] = v[i - 1] + x*deriv(v[i - 1], x) + (n - 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(11)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences

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

Related ?

Same fingerprint ?

A358623: Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (o…

References

[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