A172431 → ⊙ C(n,k+1) (shift -1) → D

nonnegative coefficients

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

Recurrence

\(\left(-24\,n - 8\,n^{2} + 58\,n^{3} - 40\,n^{4} + 8\,n^{5}\right)\,P_{n}(x) = \left(-360\,x - 6\,n + 1410\,n x + 4\,n^{2} - 2154\,n^{2} x + 18\,n^{3} + 1580\,n^{3} x - 24\,n^{4} - 548\,n^{4} x + 8\,n^{5} + 72\,n^{5} x\right)\,P_{n-1}(x) + \left(48\,x - 312\,x^{2} - 158\,n x + 1010\,n x^{2} + 190\,n^{2} x - 1202\,n^{2} x^{2} - 96\,n^{3} x + 608\,n^{3} x^{2} + 16\,n^{4} x - 104\,n^{4} x^{2}\right)\,P_{n-1}'(x) + \left(12\,x^{2} - 48\,x^{3} - 33\,n x^{2} + 132\,n x^{3} + 31\,n^{2} x^{2} - 124\,n^{2} x^{3} - 10\,n^{3} x^{2} + 40\,n^{3} x^{3}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	1							1
3	4	2						6
4	9	20	3					32
5	16	84	60	4				164
6	25	240	420	140	5			830
7	36	550	1800	1512	280	6		4184
8	49	1092	5775	9240	4410	504	7	21077

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-0c631b5705259244 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-ce89a03987d3a187

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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:  0;
   2:  1;
   3:  4,    2;
   4:  9,   20,     3;
   5: 16,   84,    60,      4;
   6: 25,  240,   420,    140,      5;
   7: 36,  550,  1800,   1512,    280,      6;
   8: 49, 1092,  5775,   9240,   4410,    504,     7;
   9: 64, 1960, 15288,  40040,  36960,  11088,   840,    8;
  10: 81, 3264, 35280, 137592, 210210, 123552, 24948, 1320, 9;

%F Formula

The row polynomials satisfy (8*n^5-40*n^4+58*n^3-8*n^2-24*n)*P(n,x) = (8*n^5-24*n^4+18*n^3+4*n^2-6*n+(72*n^5-548*n^4+1580*n^3-2154*n^2+1410*n-360)*x)*P(n-1,x) + ((16*n^4-96*n^3+190*n^2-158*n+48)*x-(104*n^4-608*n^3+1202*n^2-1010*n+312)*x^2)*d/dx P(n-1,x) - ((10*n^3-31*n^2+33*n-12)*x^2-(40*n^3-124*n^2+132*n-48)*x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^5 - 24*n^4 + 18*n^3 + 4*n^2 - 6*n + (72*n^5 - 548*n^4 + 1580*n^3 - 2154*n^2 + 1410*n - 360)*x)*P[n - 1, x] + ((16*n^4 - 96*n^3 + 190*n^2 - 158*n + 48)*x - (104*n^4 - 608*n^3 + 1202*n^2 - 1010*n + 312)*x^2)*D[P[n - 1, x], x] - ((10*n^3 - 31*n^2 + 33*n - 12)*x^2 - (40*n^3 - 124*n^2 + 132*n - 48)*x^3)*D[P[n - 1, x], {x, 2}])/(8*n^5 - 40*n^4 + 58*n^3 - 8*n^2 - 24*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^5 - 24*n^4 + 18*n^3 + 4*n^2 - 6*n + (72*n^5 - 548*n^4 + 1580*n^3 - 2154*n^2 + 1410*n - 360)*x)*v[i - 1] + ((16*n^4 - 96*n^3 + 190*n^2 - 158*n + 48)*x - (104*n^4 - 608*n^3 + 1202*n^2 - 1010*n + 312)*x^2)*deriv(v[i - 1], x) - ((10*n^3 - 31*n^2 + 33*n - 12)*x^2 - (40*n^3 - 124*n^2 + 132*n - 48)*x^3)*deriv(deriv(v[i - 1], x), x))/(8*n^5 - 40*n^4 + 58*n^3 - 8*n^2 - 24*n)); 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-2aa190e7e347a111
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-2aa190e7e347a111
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-2aa190e7e347a111

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-2aa190e7e347a111

Laguerre’s theorem on differential operators [SP14] 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