A102365 → θ+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	1	2						3
3	1	10	3					14
4	1	36	45	4				86
5	1	116	387	148	5			657
6	1	358	2631	2584	415	6		5995
7	1	1086	15840	32120	13425	1062	7	63541

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ce30cad803145faa · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-74199812e346d89c

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,    2;
  3: 1,   10,      3;
  4: 1,   36,     45,       4;
  5: 1,  116,    387,     148,       5;
  6: 1,  358,   2631,    2584,     415,       6;
  7: 1, 1086,  15840,   32120,   13425,    1062,      7;
  8: 1, 3272,  88974,  330440,  281425,   60012,   2569,    8;
  9: 1, 9832, 479226, 3024872, 4598075, 2010972, 243439, 5992, 9;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n+(2*n^2-4*n)*x-(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (2*n*x-(2*n^2-n)*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) - (1-(2*n^2-4*n+2)*x)*P(n-1,x) + ((2*n-2)*x-(n-1)*x^2)*d/dx P(n-1,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[Together[((n + (2*n^2 - 4*n)*x - (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (2*n*x - (2*n^2 - n)*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] - (1 - (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((2*n - 2)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = ((n + (2*n^2 - 4*n)*x - (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (2*n*x - (2*n^2 - n)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) - (1 - (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((2*n - 2)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-5503999bc1e6bd41
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-5503999bc1e6bd41
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-5503999bc1e6bd41
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-5503999bc1e6bd41

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image euler_polynomial, shift 0 e-5503999bc1e6bd41

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-1ce099339e80292c root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

No candidate equalities or related sequences.

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.