A056241 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 12\,x - 6\,n + 17\,n x - 13\,n x^{2} + 2\,n x^{3} - 6\,n^{2} x + 18\,n^{2} x^{2} - 6\,n^{2} x^{3} + n^{3} x - 5\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n}(x) = \left(6 - 12\,x - 54\,x^{2} - 12\,x^{3} + 6\,x^{4} - 6\,n + 17\,n x + 37\,n x^{2} - 17\,n x^{3} - 16\,n x^{4} - 6\,n^{2} x + 22\,n^{2} x^{2} + 28\,n^{2} x^{3} - 2\,n^{2} x^{4} + n^{3} x - 5\,n^{3} x^{2} + n^{3} x^{3} + 12\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-12\,x - 16\,x^{2} - 12\,x^{3} + 4\,x^{5} - 10\,n x^{2} - 18\,n x^{3} - 18\,n x^{4} - 8\,n x^{5} + 2\,n^{2} x^{2} - 6\,n^{2} x^{3} - 6\,n^{2} x^{4} - 8\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	6	3													10
3	1	12	18	4												35
4	1	20	57	40	5											123
5	1	30	135	180	75	6										427
6	1	42	270	564	450	126	7									1460
7	1	56	483	1428	1785	966	196	8								4923
8	1	72	798	3136	5535	4704	1862	288	9							16405
9	1	90	1242	6216	14535	17442	10878	3312	405	10						54131
10	1	110	1845	11400	33825	53718	47355	22800	5535	550	11					177150
11	1	132	2640	19668	71775	144408	168476	114840	44253	8800	726	12				575731
12	1	156	3663	32296	141570	349668	516523	466224	254826	80740	13431	936	13			1860047
13	1	182	4953	50908	263120	779064	1411501	1613144	1168596	526240	139997	19812	1183	14		5978715
14	1	210	6552	77532	465465	1621620	3518151	4929816	4523337	2702700	1024023	232596	28392	1470	15	19131880

Technical details

Node id: n-46287fb5b5b8ec61 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-65a1d468bad9872b

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: 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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  6,    3;
  3: 1, 12,   18,    4;
  4: 1, 20,   57,   40,     5;
  5: 1, 30,  135,  180,    75,     6;
  6: 1, 42,  270,  564,   450,   126,     7;
  7: 1, 56,  483, 1428,  1785,   966,   196,    8;
  8: 1, 72,  798, 3136,  5535,  4704,  1862,  288,   9;
  9: 1, 90, 1242, 6216, 14535, 17442, 10878, 3312, 405, 10;

%F Formula

The row polynomials satisfy -(6*n-6-(n^3-6*n^2+17*n-12)*x+(5*n^3-18*n^2+13*n)*x^2-(4*n^3-6*n^2+2*n)*x^3)*P(n,x) = -(6*n-6-(n^3-6*n^2+17*n-12)*x+(5*n^3-22*n^2-37*n+54)*x^2-(n^3+28*n^2-17*n-12)*x^3-(12*n^3-2*n^2-16*n+6)*x^4)*P(n-1,x) - (12*x-(2*n^2-10*n-16)*x^2+(6*n^2+18*n+12)*x^3+(6*n^2+18*n)*x^4+(8*n^2+8*n-4)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 6 - (n^3 - 6*n^2 + 17*n - 12)*x + (5*n^3 - 22*n^2 - 37*n + 54)*x^2 - (n^3 + 28*n^2 - 17*n - 12)*x^3 - (12*n^3 - 2*n^2 - 16*n + 6)*x^4)*P[n - 1, x] - (12*x - (2*n^2 - 10*n - 16)*x^2 + (6*n^2 + 18*n + 12)*x^3 + (6*n^2 + 18*n)*x^4 + (8*n^2 + 8*n - 4)*x^5)*D[P[n - 1, x], x])/(-6*n + 6 + (n^3 - 6*n^2 + 17*n - 12)*x - (5*n^3 - 18*n^2 + 13*n)*x^2 + (4*n^3 - 6*n^2 + 2*n)*x^3)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 6 - (n^3 - 6*n^2 + 17*n - 12)*x + (5*n^3 - 22*n^2 - 37*n + 54)*x^2 - (n^3 + 28*n^2 - 17*n - 12)*x^3 - (12*n^3 - 2*n^2 - 16*n + 6)*x^4)*v[i - 1] - (12*x - (2*n^2 - 10*n - 16)*x^2 + (6*n^2 + 18*n + 12)*x^3 + (6*n^2 + 18*n)*x^4 + (8*n^2 + 8*n - 4)*x^5)*deriv(v[i - 1], x))/(-6*n + 6 + (n^3 - 6*n^2 + 17*n - 12)*x - (5*n^3 - 18*n^2 + 13*n)*x^2 + (4*n^3 - 6*n^2 + 2*n)*x^3)); 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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
  - 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 (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.