A136011: Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) terms per row.

A136011 ↗ nonnegative coefficients

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

Recurrence from A136011

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	1							2
3	1	3							4
4	1	7	1						9
5	1	15	6						22
6	1	31	25	1					58
7	1	63	90	10					164
8	1	127	301	65	1				495
9	1	255	966	350	15				1587
10	1	511	3025	1701	140	1			5379
11	1	1023	9330	7770	1050	21			19195
12	1	2047	28501	34105	6951	266	1		71872
13	1	4095	86526	145750	42525	2646	28		281571
14	1	8191	261625	611501	246730	22827	462	1	1151338

b-files: A024427 (row sums), A136011 ?

Technical details

Node id: n-cfa0bcf0bbd4ed6d · revision 0 · JSON
Definition source: oeis_catalog:A136011
Fingerprint: f6-dd0acf0786f0fbb8

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 ?

A136011 triangle 56 terms, 14 rows · tabf · nonn · catalog b-file?

Row sums ? A024427, A171367

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,   1;
  3: 1,   3;
  4: 1,   7,   1;
  5: 1,  15,   6;
  6: 1,  31,  25,   1;
  7: 1,  63,  90,  10;
  8: 1, 127, 301,  65,  1;
  9: 1, 255, 966, 350, 15;

%F Formula

The row polynomials satisfy P(n,x) = P(n-1,x) + x*d/dx P(n-1,x) + 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] + 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) + 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

A136011 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1;
   3: 1,   1;
   4: 1,   3;
   5: 1,   7,   1;
   6: 1,  15,   6;
   7: 1,  31,  25,   1;
   8: 1,  63,  90,  10;
   9: 1, 127, 301,  65,  1;
  10: 1, 255, 966, 350, 15;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + x*D[P[n - 1, x], x] + x*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] = 1; for(n = 3, N, my(i = n); v[i] = v[i - 1] + x*deriv(v[i - 1], x) + 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, v[i])))};
T(10)

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

Used in 5

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