A152842 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-9 - 15\,x - x^{2} + 4\,x^{3} + x^{4} - n + 9\,n x + 9\,n x^{2} + 2\,n x^{3} + 4\,n^{2} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-45 - 96\,x - 9\,x^{2} + 45\,x^{3} + 22\,x^{4} + 3\,x^{5} - 3\,n + 96\,n x + 90\,n x^{2} + 13\,n x^{3} - 8\,n x^{4} - 2\,n x^{5} + 40\,n^{2} + 10\,n^{2} x - 35\,n^{2} x^{2} - 23\,n^{2} x^{3} - 4\,n^{2} x^{4} - 16\,n^{3} - 24\,n^{3} x - 12\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-9 - 6\,x + 62\,x^{2} + 114\,x^{3} + 73\,x^{4} + 20\,x^{5} + 2\,x^{6} + 12\,n + 94\,n x + 168\,n x^{2} + 118\,n x^{3} + 36\,n x^{4} + 4\,n x^{5} + 24\,n^{2} + 56\,n^{2} x + 46\,n^{2} x^{2} + 16\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	1														5
2	7	6	1													14
3	24	29	10	1												64
4	33	68	46	12	1											160
5	108	261	228	90	16	1										704
6	135	450	579	364	117	18	1									1664
7	432	1593	2358	1807	772	183	22	1								7168
8	513	2376	4572	4744	2886	1048	220	24	1							16384
9	1620	8073	17064	20004	14288	6430	1816	308	28	1						69632
10	1863	11070	28485	41640	38150	22804	8970	2280	355	30	1					155648
11	5832	36693	101250	161325	164520	112506	52524	16690	3520	465	34	1				655360
12	6561	47628	153738	291060	359055	303288	179404	74664	21615	4220	522	36	1			1441792
13	20412	155277	529740	1071198	1429920	1328391	882136	423220	146364	35915	6044	654	40	1		6029312
14	22599	193914	753543	1754676	2730483	2998422	2391907	1405464	609973	194054	44373	7028	721	42	1	13107200

Technical details

Node id: n-2732342fe6245c08 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b818def587ffa8ed

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:    4,    1;
  2:    7,    6,     1;
  3:   24,   29,    10,     1;
  4:   33,   68,    46,    12,     1;
  5:  108,  261,   228,    90,    16,    1;
  6:  135,  450,   579,   364,   117,   18,    1;
  7:  432, 1593,  2358,  1807,   772,  183,   22,   1;
  8:  513, 2376,  4572,  4744,  2886, 1048,  220,  24,  1;
  9: 1620, 8073, 17064, 20004, 14288, 6430, 1816, 308, 28, 1;

%F Formula

The row polynomials satisfy (4*n^2-n-9+(4*n^2+9*n-15)*x+(n^2+9*n-1)*x^2+(2*n+4)*x^3+x^4)*P(n,x) = -(16*n^3-40*n^2+3*n+45+(24*n^3-10*n^2-96*n+96)*x+(12*n^3+35*n^2-90*n+9)*x^2+(2*n^3+23*n^2-13*n-45)*x^3+(4*n^2+8*n-22)*x^4+(2*n-3)*x^5)*P(n-1,x) + (24*n^2+12*n-9+(56*n^2+94*n-6)*x+(46*n^2+168*n+62)*x^2+(16*n^2+118*n+114)*x^3+(2*n^2+36*n+73)*x^4+(4*n+20)*x^5+2*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n^3 - 40*n^2 + 3*n + 45 + (24*n^3 - 10*n^2 - 96*n + 96)*x + (12*n^3 + 35*n^2 - 90*n + 9)*x^2 + (2*n^3 + 23*n^2 - 13*n - 45)*x^3 + (4*n^2 + 8*n - 22)*x^4 + (2*n - 3)*x^5)*P[n - 1, x] + (24*n^2 + 12*n - 9 + (56*n^2 + 94*n - 6)*x + (46*n^2 + 168*n + 62)*x^2 + (16*n^2 + 118*n + 114)*x^3 + (2*n^2 + 36*n + 73)*x^4 + (4*n + 20)*x^5 + 2*x^6)*D[P[n - 1, x], x])/(4*n^2 - n - 9 + (4*n^2 + 9*n - 15)*x + (n^2 + 9*n - 1)*x^2 + (2*n + 4)*x^3 + x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(16*n^3 - 40*n^2 + 3*n + 45 + (24*n^3 - 10*n^2 - 96*n + 96)*x + (12*n^3 + 35*n^2 - 90*n + 9)*x^2 + (2*n^3 + 23*n^2 - 13*n - 45)*x^3 + (4*n^2 + 8*n - 22)*x^4 + (2*n - 3)*x^5)*v[i - 1] + (24*n^2 + 12*n - 9 + (56*n^2 + 94*n - 6)*x + (46*n^2 + 168*n + 62)*x^2 + (16*n^2 + 118*n + 114)*x^3 + (2*n^2 + 36*n + 73)*x^4 + (4*n + 20)*x^5 + 2*x^6)*deriv(v[i - 1], x))/(4*n^2 - n - 9 + (4*n^2 + 9*n - 15)*x + (n^2 + 9*n - 1)*x^2 + (2*n + 4)*x^3 + x^4)); 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 reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,..…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,..…?
    no recorded property covers this requirement
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.