A056241: Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

A056241 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A056241

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	3	1						5
3	1	6	6	1					14
4	1	10	19	10	1				41
5	1	15	45	45	15	1			122
6	1	21	90	141	90	21	1		365
7	1	28	161	357	357	161	28	1	1094

All 15 rows b-files: A007051 (row sums), A056241, A056241 (reversed triangle) ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-2 - 6\,x^{2} + 2\,n - n x + 10\,n x^{2} + n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(2\,x - 6\,x^{2} - 14\,x^{3} + 4\,n x^{2} + 8\,n x^{3}\right)\,P_{n-1}'(x) + \left(-4\,x^{2} - 4\,x^{3} - 4\,x^{4}\right)\,P_{n-1}''(x)\)

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

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

Technical details

Node id: n-a6f3b35778192ffb · revision 1 · JSON
Definition source: oeis_catalog:A056241
derived:reversal (equal by e-d32fa4f50d269032)
Fingerprint: f6-90ff47d73675b1af

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A056241 reversed triangle 65 terms, 10 rows · tabl · nonn · catalog b-file?
A056241 triangle 65 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A007051, A124302

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   1;
  3: 1,  6,   6,    1;
  4: 1, 10,  19,   10,    1;
  5: 1, 15,  45,   45,   15,    1;
  6: 1, 21,  90,  141,   90,   21,    1;
  7: 1, 28, 161,  357,  357,  161,   28,   1;
  8: 1, 36, 266,  784, 1107,  784,  266,  36,  1;
  9: 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1;

%F Formula

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

%t Mathematica

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

A056241 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  3,   1;
   4: 1,  6,   6,    1;
   5: 1, 10,  19,   10,    1;
   6: 1, 15,  45,   45,   15,    1;
   7: 1, 21,  90,  141,   90,   21,    1;
   8: 1, 28, 161,  357,  357,  161,   28,   1;
   9: 1, 36, 266,  784, 1107,  784,  266,  36,  1;
  10: 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 1] - (1 + x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A056241 reversed triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  3,   1;
   4: 1,  6,   6,    1;
   5: 1, 10,  19,   10,    1;
   6: 1, 15,  45,   45,   15,    1;
   7: 1, 21,  90,  141,   90,   21,    1;
   8: 1, 28, 161,  357,  357,  161,   28,   1;
   9: 1, 36, 266,  784, 1107,  784,  266,  36,  1;
  10: 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1+x+x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x + x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 1] - (1 + x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); 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 Affine, reversal and Möbius substitutions (elementary), Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), 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
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
↑ Liu–Wang theorem for three-term recurrences (above)

Used in 12

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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