A026374 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 3 + 2\,x\)

\(P_{2}(x) = 7 + 12\,x + 3\,x^{2}\)

\(P_{3}(x) = 9 + 32\,x + 24\,x^{2} + 4\,x^{3}\)

\(P_{4}(x) = 13 + 78\,x + 105\,x^{2} + 44\,x^{3} + 5\,x^{4}\)

\(P_{5}(x) = 15 + 116\,x + 255\,x^{2} + 208\,x^{3} + 65\,x^{4} + 6\,x^{5}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	7	12	3													22
3	9	32	24	4												69
4	13	78	105	44	5											245
5	15	116	255	208	65	6										665
6	19	198	630	780	420	96	7									2150
7	21	254	1017	1800	1545	654	126	8								5425
8	25	372	1902	4476	5295	3300	1078	168	9							16625
9	27	446	2634	7588	11865	10518	5334	1496	207	10						40125
10	31	600	4245	15020	29615	34008	23275	9480	2205	260	11					118750
11	33	692	5430	22000	51815	74784	67746	38720	13725	2860	308	12				278125
12	37	882	7983	37920	106590	187086	210371	153504	72630	21810	3927	372	13			803125
13	39	992	9729	51084	162840	334542	456561	419992	261252	108810	29469	4872	429	14		1840625
14	43	1218	13440	80304	294945	707574	1143779	1266240	967617	510090	183183	43428	6370	504	15	5218750

Technical details

Node id: n-5eddde11117f58a0 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4d9e8666b18989c5

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:  1;
  1:  3,   2;
  2:  7,  12,    3;
  3:  9,  32,   24,    4;
  4: 13,  78,  105,   44,     5;
  5: 15, 116,  255,  208,    65,     6;
  6: 19, 198,  630,  780,   420,    96,    7;
  7: 21, 254, 1017, 1800,  1545,   654,  126,    8;
  8: 25, 372, 1902, 4476,  5295,  3300, 1078,  168,   9;
  9: 27, 446, 2634, 7588, 11865, 10518, 5334, 1496, 207, 10;

%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) = 1, P(1,x) = 3 + 2*x, P(2,x) = 7 + 12*x + 3*x^2, P(3,x) = 9 + 32*x + 24*x^2 + 4*x^3, P(4,x) = 13 + 78*x + 105*x^2 + 44*x^3 + 5*x^4, P(5,x) = 15 + 116*x + 255*x^2 + 208*x^3 + 65*x^4 + 6*x^5.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 7 + 12*x + 3*x^2;
P[3, x_] := 9 + 32*x + 24*x^2 + 4*x^3;
P[4, x_] := 13 + 78*x + 105*x^2 + 44*x^3 + 5*x^4;
P[5, x_] := 15 + 116*x + 255*x^2 + 208*x^3 + 65*x^4 + 6*x^5;
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] = 1; v[2] = 3 + 2*x; v[3] = 7 + 12*x + 3*x^2; v[4] = 9 + 32*x + 24*x^2 + 4*x^3; v[5] = 13 + 78*x + 105*x^2 + 44*x^3 + 5*x^4; v[6] = 15 + 116*x + 255*x^2 + 208*x^3 + 65*x^4 + 6*x^5; 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 theta_plus_one 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,k) f…?
  - Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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.