A172431 → ⊙ C(n,k+1) (shift -1) → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 8\,n - 10\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(-2 - 52\,x + 8\,n + 152\,n x - 10\,n^{2} - 132\,n^{2} x + 4\,n^{3} + 36\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 62\,x^{2} - 18\,n x + 134\,n x^{2} + 8\,n^{2} x - 52\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(3\,x^{2} - 12\,x^{3} - 5\,n x^{2} + 20\,n x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	2							3
3	1	8	3						12
4	1	18	30	4					53
5	1	32	126	80	5				244
6	1	50	360	560	175	6			1152
7	1	72	825	2400	1890	336	7		5531
8	1	98	1638	7700	11550	5292	588	8	26875

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-012ae01d9db4fd27 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c6c82bb93c30543f

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 -1)
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 ?

A262442: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,   2;
   3: 1,   8,    3;
   4: 1,  18,   30,     4;
   5: 1,  32,  126,    80,      5;
   6: 1,  50,  360,   560,    175,      6;
   7: 1,  72,  825,  2400,   1890,    336,      7;
   8: 1,  98, 1638,  7700,  11550,   5292,    588,     8;
   9: 1, 128, 2940, 20384,  50050,  44352,  12936,   960,    9;
  10: 1, 162, 4896, 47040, 171990, 252252, 144144, 28512, 1485, 10;

%F Formula

The row polynomials satisfy (4*n^3-10*n^2+8*n-2)*P(n,x) = (4*n^3-10*n^2+8*n-2+(36*n^3-132*n^2+152*n-52)*x)*P(n-1,x) + ((8*n^2-18*n+8)*x-(52*n^2-134*n+62)*x^2)*d/dx P(n-1,x) - ((5*n-3)*x^2-(20*n-12)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 10*n^2 + 8*n - 2 + (36*n^3 - 132*n^2 + 152*n - 52)*x)*P[n - 1, x] + ((8*n^2 - 18*n + 8)*x - (52*n^2 - 134*n + 62)*x^2)*D[P[n - 1, x], x] - ((5*n - 3)*x^2 - (20*n - 12)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 10*n^2 + 8*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^3 - 10*n^2 + 8*n - 2 + (36*n^3 - 132*n^2 + 152*n - 52)*x)*v[i - 1] + ((8*n^2 - 18*n + 8)*x - (52*n^2 - 134*n + 62)*x^2)*deriv(v[i - 1], x) - ((5*n - 3)*x^2 - (20*n - 12)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 10*n^2 + 8*n - 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-fe1ef6d74823af06
Coefficients of one sign. Every nonzero row, n >= 1: 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-fe1ef6d74823af06
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-fe1ef6d74823af06

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-fe1ef6d74823af06

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image hadamard_multiplier, shift -1 e-1491609a6fc5f8c9
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): Even row Pascal-square read by antidiagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-83282836b5ac538e root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023