A059344 → ⊙ 1/k! (shift -1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 15\,x - 4\,x^{2} + 12\,n - 19\,n x + 4\,n x^{2} - 6\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-3 - 17\,x + 84\,x^{2} - 32\,x^{3} + 6\,n + 37\,n x - 72\,n x^{2} + 16\,n x^{3} - 4\,n^{2} - 24\,n^{2} x + 16\,n^{2} x^{2} + n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(4\,x - 30\,x^{2} + 64\,x^{3} - 32\,x^{4} - 6\,n x + 32\,n x^{2} - 32\,n x^{3} + 2\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
1	1				1
2	1				1
3	3	2			5
4	7	6			13
5	13	24	6		43
6	21	80	30		131
7	31	210	150	20	411
8	43	462	630	140	1275

Show all 15 stored rows

Generated b-files: A391990 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-f88ecd1d5f2ce358 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4806a24fb0d130c1

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 ?

A391990: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:  3,    2;
   4:  7,    6;
   5: 13,   24,    6;
   6: 21,   80,   30;
   7: 31,  210,  150,   20;
   8: 43,  462,  630,  140;
   9: 57,  896, 2100,  840,  70;
  10: 73, 1584, 5796, 4200, 630;

%F Formula

The row polynomials satisfy (n^3-6*n^2+12*n-7+(4*n^2-19*n+15)*x+(4*n-4)*x^2)*P(n,x) = (n^3-4*n^2+6*n-3+(4*n^3-24*n^2+37*n-17)*x+(16*n^2-72*n+84)*x^2+(16*n-32)*x^3)*P(n-1,x) + ((2*n^2-6*n+4)*x-(8*n^2-32*n+30)*x^2-(32*n-64)*x^3-32*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^3 - 4*n^2 + 6*n - 3 + (4*n^3 - 24*n^2 + 37*n - 17)*x + (16*n^2 - 72*n + 84)*x^2 + (16*n - 32)*x^3)*P[n - 1, x] + ((2*n^2 - 6*n + 4)*x - (8*n^2 - 32*n + 30)*x^2 - (32*n - 64)*x^3 - 32*x^4)*D[P[n - 1, x], x])/(n^3 - 6*n^2 + 12*n - 7 + (4*n^2 - 19*n + 15)*x + (4*n - 4)*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^3 - 4*n^2 + 6*n - 3 + (4*n^3 - 24*n^2 + 37*n - 17)*x + (16*n^2 - 72*n + 84)*x^2 + (16*n - 32)*x^3)*v[i - 1] + ((2*n^2 - 6*n + 4)*x - (8*n^2 - 32*n + 30)*x^2 - (32*n - 64)*x^3 - 32*x^4)*deriv(v[i - 1], x))/(n^3 - 6*n^2 + 12*n - 7 + (4*n^2 - 19*n + 15)*x + (4*n - 4)*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-272eb8130991447b
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-272eb8130991447b
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-272eb8130991447b

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 derivative_polynomial, shift 0 e-272eb8130991447b

Laguerre’s theorem on differential operators [SP14] 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) → D+1

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.