A059344 → ⊙ 1/k! (shift -1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n - 10\,n x + 12\,n^{2} x - n^{2} x^{2} - 4\,n^{3} x + 3\,n^{3} x^{2} - 3\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-1 - 30\,x - 32\,x^{2} + n + 6\,n x + 8\,n x^{2} + 12\,n^{2} x + 59\,n^{2} x^{2} - 8\,n^{2} x^{3} - 4\,n^{3} x - 25\,n^{3} x^{2} - 12\,n^{3} x^{3} - 3\,n^{4} x^{2} + n^{5} x^{2} + 4\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} - 16\,x^{3} - 8\,n^{2} x^{2} + 34\,n^{2} x^{3} - 8\,n^{2} x^{4} + 4\,n^{3} x^{3} - 16\,n^{3} x^{4} + 2\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
1	1				1
2	1				1
3	1	8			9
4	1	24			25
5	1	48	54		103
6	1	80	270		351
7	1	120	810	320	1251
8	1	168	1890	2240	4299

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-15c46876416f2009 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ab1b08e67af73826

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 -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 ?

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1;
   3: 1,   8;
   4: 1,  24;
   5: 1,  48,   54;
   6: 1,  80,  270;
   7: 1, 120,  810,   320;
   8: 1, 168, 1890,  2240;
   9: 1, 224, 3780,  8960,  1750;
  10: 1, 288, 6804, 26880, 15750;

%F Formula

The row polynomials satisfy (n-1-(4*n^3-12*n^2+10*n-2)*x+(n^5-3*n^4+3*n^3-n^2)*x^2)*P(n,x) = (n-1-(4*n^3-12*n^2-6*n+30)*x+(n^5-3*n^4-25*n^3+59*n^2+8*n-32)*x^2+(4*n^5-12*n^3-8*n^2)*x^3)*P(n-1,x) + (2*x-(8*n^2+4)*x^2+(2*n^4+4*n^3+34*n^2-16)*x^3-(8*n^4+16*n^3+8*n^2)*x^4)*d/dx 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[((n - 1 - (4*n^3 - 12*n^2 - 6*n + 30)*x + (n^5 - 3*n^4 - 25*n^3 + 59*n^2 + 8*n - 32)*x^2 + (4*n^5 - 12*n^3 - 8*n^2)*x^3)*P[n - 1, x] + (2*x - (8*n^2 + 4)*x^2 + (2*n^4 + 4*n^3 + 34*n^2 - 16)*x^3 - (8*n^4 + 16*n^3 + 8*n^2)*x^4)*D[P[n - 1, x], x])/(n - 1 - (4*n^3 - 12*n^2 + 10*n - 2)*x + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^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] = ((n - 1 - (4*n^3 - 12*n^2 - 6*n + 30)*x + (n^5 - 3*n^4 - 25*n^3 + 59*n^2 + 8*n - 32)*x^2 + (4*n^5 - 12*n^3 - 8*n^2)*x^3)*v[i - 1] + (2*x - (8*n^2 + 4)*x^2 + (2*n^4 + 4*n^3 + 34*n^2 - 16)*x^3 - (8*n^4 + 16*n^3 + 8*n^2)*x^4)*deriv(v[i - 1], x))/(n - 1 - (4*n^3 - 12*n^2 + 10*n - 2)*x + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^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-23d8897c84d2236f
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-23d8897c84d2236f
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-23d8897c84d2236f

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-23d8897c84d2236f

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.
  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-4d3032ecd7423900
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers paper_theorem e-94e95c2ea0369f05 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A011973 → ⊙ C(n,k) → (θ+1)²

References

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