A056242 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6\,x^{2} - 4\,n + n x + 16\,n x^{2} + 4\,n^{2} - 4\,n^{2} x - 14\,n^{2} x^{2} + 3\,n^{3} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(30\,x^{2} - 8\,n - 38\,n x - 39\,n x^{2} + 22\,n x^{3} + 8\,n^{2} + 16\,n^{2} x - 16\,n^{2} x^{2} - 34\,n^{2} x^{3} + 6\,n^{3} x + 17\,n^{3} x^{2} + 12\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(12\,x^{2} + 12\,x^{3} - 8\,n x - 18\,n x^{2} - 2\,n x^{3} + 8\,n x^{4} - 6\,n^{2} x^{2} - 14\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	2							4
3	4	10	3						17
4	8	32	27	4					71
5	16	88	123	56	5				288
6	32	224	438	340	100	6			1140
7	64	544	1368	1508	775	162	7		4428
8	128	1280	3936	5632	4165	1554	245	8	16948

All 15 rows

Technical details

Node id: n-ac2000756e296595 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-eec6c58ba17db0cb

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   2,    2;
   3:   4,   10,     3;
   4:   8,   32,    27,     4;
   5:  16,   88,   123,    56,     5;
   6:  32,  224,   438,   340,   100,     6;
   7:  64,  544,  1368,  1508,   775,   162,     7;
   8: 128, 1280,  3936,  5632,  4165,  1554,   245,    8;
   9: 256, 2944, 10704, 18848, 18245,  9912,  2842,  352,   9;
  10: 512, 6656, 27936, 58432, 70010, 50166, 21168, 4848, 486, 10;

%F Formula

The row polynomials satisfy (4*n^2-4*n+(3*n^3-4*n^2+n)*x+(4*n^3-14*n^2+16*n-6)*x^2)*P(n,x) = (8*n^2-8*n+(6*n^3+16*n^2-38*n)*x+(17*n^3-16*n^2-39*n+30)*x^2+(12*n^3-34*n^2+22*n)*x^3)*P(n-1,x) - (8*n*x+(6*n^2+18*n-12)*x^2+(14*n^2+2*n-12)*x^3+(8*n^2-8*n)*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[((8*n^2 - 8*n + (6*n^3 + 16*n^2 - 38*n)*x + (17*n^3 - 16*n^2 - 39*n + 30)*x^2 + (12*n^3 - 34*n^2 + 22*n)*x^3)*P[n - 1, x] - (8*n*x + (6*n^2 + 18*n - 12)*x^2 + (14*n^2 + 2*n - 12)*x^3 + (8*n^2 - 8*n)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 4*n + (3*n^3 - 4*n^2 + n)*x + (4*n^3 - 14*n^2 + 16*n - 6)*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] = ((8*n^2 - 8*n + (6*n^3 + 16*n^2 - 38*n)*x + (17*n^3 - 16*n^2 - 39*n + 30)*x^2 + (12*n^3 - 34*n^2 + 22*n)*x^3)*v[i - 1] - (8*n*x + (6*n^2 + 18*n - 12)*x^2 + (14*n^2 + 2*n - 12)*x^3 + (8*n^2 - 8*n)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 4*n + (3*n^3 - 4*n^2 + n)*x + (4*n^3 - 14*n^2 + 16*n - 6)*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    no recorded property covers this requirement
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    ↑ Affine-current basis map theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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.