A091044 → ⊙ C(n+2,k+1) → γ

nonnegative coefficients

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

Recurrence

\(\left(6 + 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(4 - 12\,x + 13\,n - 24\,n x - 2\,n^{2} + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(6\,x - 24\,x^{2} + 18\,n x - 72\,n x^{2}\right)\,P_{n-1}'(x) + \left(2 - 6\,n + 4\,n^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	1	3			4
3	1	11			12
4	1	26	26		53
5	1	50	170		221
6	1	85	646	323	1055
7	1	133	1862	3059	5055

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-6cf134ccfa9a5641 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-aaa7f30b59ecd9f9

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 0)
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,   3;
  3: 1,  11;
  4: 1,  26,   26;
  5: 1,  50,  170;
  6: 1,  85,  646,   323;
  7: 1, 133, 1862,  3059;
  8: 1, 196, 4508, 16100,  4830;
  9: 1, 276, 9660, 62100, 60030;

%F Formula

The row polynomials satisfy (2*n^2+7*n+6)*P(n,x) = -(2*n^2-13*n-4-(36*n^2-24*n-12)*x)*P(n-1,x) + ((18*n+6)*x-(72*n+24)*x^2)*d/dx P(n-1,x) + (4*n^2-6*n+2)*P(n-2,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[(-(2*n^2 - 13*n - 4 - (36*n^2 - 24*n - 12)*x)*P[n - 1, x] + ((18*n + 6)*x - (72*n + 24)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 6*n + 2)*P[n - 2, x])/(2*n^2 + 7*n + 6)]];
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] = (-(2*n^2 - 13*n - 4 - (36*n^2 - 24*n - 12)*x)*v[i - 1] + ((18*n + 6)*x - (72*n + 24)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 6*n + 2)*v[i - 2])/(2*n^2 + 7*n + 6)); 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-10ee054377433864
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-10ee054377433864
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-10ee054377433864

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

gamma_polynomial scale 1, shift 0 e-10ee054377433864

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image hadamard_multiplier, shift 0 e-00224510dcd0cfb5
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
    
    preserver_image reversal, shift 0 e-75f779345b39f910
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)
  - preserver_image reversal, shift 0 e-3a2967256f8a0d67
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ preserver_image hadamard_multiplier, shift 0 e-00224510dcd0cfb5 (expanded above)

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