A026374 → D+1 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(1 + 9\,x + 30\,x^{2} + 45\,x^{3} + 30\,x^{4} + 9\,x^{5} + x^{6}\right)\,P_{n-6}(x) + \left(-3 - 18\,x - 33\,x^{2} - 18\,x^{3} - 3\,x^{4}\right)\,P_{n-4}(x) + \left(3 + 9\,x + 3\,x^{2}\right)\,P_{n-2}(x)\)

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

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

\(P_{2}(x) = 5 + 2\,x\)

\(P_{3}(x) = 12 + 14\,x + 3\,x^{2}\)

\(P_{4}(x) = 28 + 58\,x + 30\,x^{2} + 4\,x^{3}\)

\(P_{5}(x) = 41 + 136\,x + 135\,x^{2} + 48\,x^{3} + 5\,x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	5	2						7
3	12	14	3					29
4	28	58	30	4				120
5	41	136	135	48	5			365
6	69	330	495	300	75	6		1275
7	88	528	1125	1080	495	102	7	3425

All 15 rows

Technical details

Node id: n-f4ebb65dc61ac224 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-46318af6efa716ef

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 -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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   0;
  1:   1;
  2:   5,    2;
  3:  12,   14,    3;
  4:  28,   58,   30,    4;
  5:  41,  136,  135,   48,    5;
  6:  69,  330,  495,  300,   75,    6;
  7:  88,  528, 1125, 1080,  495,  102,    7;
  8: 128,  980, 2772, 3660, 2460,  852,  140,   8;
  9: 153, 1352, 4674, 8136, 7755, 4152, 1218, 176, 9;

%F Formula

The row polynomials satisfy P(n,x) = (1+9*x+30*x^2+45*x^3+30*x^4+9*x^5+x^6)*P(n-6,x) - (3+18*x+33*x^2+18*x^3+3*x^4)*P(n-4,x) + (3+9*x+3*x^2)*P(n-2,x) for n >= 6, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x, P(3,x) = 12 + 14*x + 3*x^2, P(4,x) = 28 + 58*x + 30*x^2 + 4*x^3, P(5,x) = 41 + 136*x + 135*x^2 + 48*x^3 + 5*x^4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[3, x_] := 12 + 14*x + 3*x^2;
P[4, x_] := 28 + 58*x + 30*x^2 + 4*x^3;
P[5, x_] := 41 + 136*x + 135*x^2 + 48*x^3 + 5*x^4;
P[n_, x_] := P[n, x] = Expand[(1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*P[n - 6, x] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*P[n - 4, x] + (3 + 9*x + 3*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 5 + 2*x; v[4] = 12 + 14*x + 3*x^2; v[5] = 28 + 58*x + 30*x^2 + 4*x^3; v[6] = 41 + 136*x + 135*x^2 + 48*x^3 + 5*x^4; for(n = 6, N, my(i = n + 1); v[i] = (1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*v[i - 6] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*v[i - 4] + (3 + 9*x + 3*x^2)*v[i - 2]); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from one_plus_derivative of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1…?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.