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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
9	256	2944	10704	18848	18245	9912	2842	352	9							64112
10	512	6656	27936	58432	70010	50166	21168	4848	486	10						240224
11	1024	14848	70656	171200	244700	218190	122283	41616	7830	650	11					893008
12	2048	32768	174336	480256	797800	852000	597135	271656	76626	12100	847	12				3297584
13	4096	71680	421632	1301504	2464480	3068280	2585135	1477520	560439	133760	18029	1080	13			12107648
14	8192	155648	1003008	3429376	7291840	10373568	10218670	7045480	3371940	1088230	223366	26052	1352	14		44236736
15	16384	335872	2353152	8827904	20827520	33337920	37612400	30402440	17610255	7200050	2008721	359268	36673	1666	15	160930240

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.