Veronese parent of A109447 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2} + 3\,n + 2\,n x + n^{2}\right)\,P_{n}(x) = \left(1 + 5\,x + 8\,x^{2} + 5\,x^{3} + x^{4} + 2\,n + 7\,n x + 10\,n x^{2} + 6\,n x^{3} + n x^{4} - 2\,n^{2} + n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3} - n^{3} + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(1 - x - 9\,x^{2} - 12\,x^{3} - 6\,x^{4} - x^{5} + 3\,n + 3\,n x - 5\,n x^{2} - 7\,n x^{3} - 2\,n x^{4} + n^{2} + n^{2} x - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 3\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	3	1															5
1	3	4	1															8
2	1	7	12	7	1													28
3	5	16	18	8	1													48
4	1	11	35	50	35	11	1											144
5	7	36	75	80	45	12	1											256
6	1	15	70	161	210	161	70	15	1									704
7	9	64	196	336	350	224	84	16	1									1280
8	1	19	117	372	714	882	714	372	117	19	1							3328
9	11	100	405	960	1470	1512	1050	480	135	20	1							6144
10	1	23	176	715	1815	3102	3696	3102	1815	715	176	23	1					15360
11	13	144	726	2200	4455	6336	6468	4752	2475	880	198	24	1					28672
12	1	27	247	1222	3861	8437	13299	15444	13299	8437	3861	1222	247	27	1			69632
13	15	196	1183	4368	11011	20020	27027	27456	21021	12012	5005	1456	273	28	1			131072
14	1	31	330	1925	7280	19383	38038	56485	64350	56485	38038	19383	7280	1925	330	31	1	311296

Technical details

Node id: n-5f93f478467a1f33 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b415c280d34402f4

Observed on the stored rows ?

Degrees: 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16 (irregular)
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,   3,   1;
  1:  3,   4,   1;
  2:  1,   7,  12,   7,    1;
  3:  5,  16,  18,   8,    1;
  4:  1,  11,  35,  50,   35,   11,    1;
  5:  7,  36,  75,  80,   45,   12,    1;
  6:  1,  15,  70, 161,  210,  161,   70,  15,   1;
  7:  9,  64, 196, 336,  350,  224,   84,  16,   1;
  8:  1,  19, 117, 372,  714,  882,  714, 372, 117, 19, 1;
  9: 11, 100, 405, 960, 1470, 1512, 1050, 480, 135, 20, 1;

%F Formula

The row polynomials satisfy (n^2+3*n+1+(2*n+2)*x+x^2)*P(n,x) = -(n^3+2*n^2-2*n-1-(n^2+7*n+5)*x-(n^3+5*n^2+10*n+8)*x^2-(2*n^2+6*n+5)*x^3-(n+1)*x^4)*P(n-1,x) + (n^2+3*n+1+(n^2+3*n-1)*x-(n^2+5*n+9)*x^2-(n^2+7*n+12)*x^3-(2*n+6)*x^4-x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x + x^2.

%t Mathematica

P[0, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 2*n^2 - 2*n - 1 - (n^2 + 7*n + 5)*x - (n^3 + 5*n^2 + 10*n + 8)*x^2 - (2*n^2 + 6*n + 5)*x^3 - (n + 1)*x^4)*P[n - 1, x] + (n^2 + 3*n + 1 + (n^2 + 3*n - 1)*x - (n^2 + 5*n + 9)*x^2 - (n^2 + 7*n + 12)*x^3 - (2*n + 6)*x^4 - x^5)*D[P[n - 1, x], x])/(n^2 + 3*n + 1 + (2*n + 2)*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 3*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 + 2*n^2 - 2*n - 1 - (n^2 + 7*n + 5)*x - (n^3 + 5*n^2 + 10*n + 8)*x^2 - (2*n^2 + 6*n + 5)*x^3 - (n + 1)*x^4)*v[i - 1] + (n^2 + 3*n + 1 + (n^2 + 3*n - 1)*x - (n^2 + 5*n + 9)*x^2 - (n^2 + 7*n + 12)*x^3 - (2*n + 6)*x^4 - x^5)*deriv(v[i - 1], x))/(n^2 + 3*n + 1 + (2*n + 2)*x + x^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 veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.?
  - 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

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.