A166972 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 8\,x + n\right)\,P_{n}(x) = \left(56 - 183\,x + 248\,x^{2} - 156\,x^{3} + 32\,x^{4} - 8\,n + 31\,n x - 123\,n x^{2} + 136\,n x^{3} - 32\,n x^{4} - 2\,n^{2} x + 20\,n^{2} x^{2} - 12\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(91\,x - 671\,x^{2} + 900\,x^{3} - 316\,x^{4} + 32\,x^{5} - 13\,n x + 151\,n x^{2} - 123\,n x^{3} + 12\,n x^{4} - 10\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(21\,x^{2} - 192\,x^{3} + 276\,x^{4} - 96\,x^{5} - 3\,n x^{2} + 45\,n x^{3} - 36\,n x^{4} - 3\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-71 + 192\,x - 156\,x^{2} + 32\,x^{3} + 9\,n - 31\,n x + 20\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-104\,x + 112\,x^{2} - 8\,x^{3} + 13\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(-24\,x^{2} + 24\,x^{3} + 3\,n x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	11	12	1						24
3	84	165	44	1					294
4	669	2888	1476	126	1				5160
5	5350	57313	50612	9446	314	1			123036
6	42799	1187532	1869351	611188	48285	720	1		3759876
7	342392	24895381	72741360	38698103	5649232	212055	1568	1	142540092

All 15 rows

Technical details

Node id: n-2c966c54b2b9440a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-7c049e635320e1d5

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:        2,           1;
  2:       11,          12,            1;
  3:       84,         165,           44,            1;
  4:      669,        2888,         1476,          126,           1;
  5:     5350,       57313,        50612,         9446,         314,          1;
  6:    42799,     1187532,      1869351,       611188,       48285,        720,         1;
  7:   342392,    24895381,     72741360,     38698103,     5649232,     212055,      1568,       1;
  8:  2739137,   523145400,   2899774642,   2488260890,   592107390,   42895372,    838766,    3306,    1;
  9: 21913098, 10994270817, 116533305032, 162678522604, 59708846880, 7221572330, 281683024, 3082056, 6830, 1;

%F Formula

The row polynomials satisfy (n-8+8*x)*P(n,x) = -(8*n-56+(2*n^2-31*n+183)*x+(n^3-20*n^2+123*n-248)*x^2+(12*n^2-136*n+156)*x^3+(32*n-32)*x^4)*P(n-2,x) - ((13*n-91)*x+(10*n^2-151*n+671)*x^2-(n^2-123*n+900)*x^3-(12*n-316)*x^4-32*x^5)*d/dx P(n-2,x) - ((3*n-21)*x^2+(3*n^2-45*n+192)*x^3+(36*n-276)*x^4+96*x^5)*d^2/dx^2 P(n-2,x) + (9*n-71+(2*n^2-31*n+192)*x+(20*n-156)*x^2+32*x^3)*P(n-1,x) + ((13*n-104)*x-(n-112)*x^2-8*x^3)*d/dx P(n-1,x) + ((3*n-24)*x^2+24*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 56 + (2*n^2 - 31*n + 183)*x + (n^3 - 20*n^2 + 123*n - 248)*x^2 + (12*n^2 - 136*n + 156)*x^3 + (32*n - 32)*x^4)*P[n - 2, x] - ((13*n - 91)*x + (10*n^2 - 151*n + 671)*x^2 - (n^2 - 123*n + 900)*x^3 - (12*n - 316)*x^4 - 32*x^5)*D[P[n - 2, x], x] - ((3*n - 21)*x^2 + (3*n^2 - 45*n + 192)*x^3 + (36*n - 276)*x^4 + 96*x^5)*D[P[n - 2, x], {x, 2}] + (9*n - 71 + (2*n^2 - 31*n + 192)*x + (20*n - 156)*x^2 + 32*x^3)*P[n - 1, x] + ((13*n - 104)*x - (n - 112)*x^2 - 8*x^3)*D[P[n - 1, x], x] + ((3*n - 24)*x^2 + 24*x^3)*D[P[n - 1, x], {x, 2}])/(n - 8 + 8*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n - 56 + (2*n^2 - 31*n + 183)*x + (n^3 - 20*n^2 + 123*n - 248)*x^2 + (12*n^2 - 136*n + 156)*x^3 + (32*n - 32)*x^4)*v[i - 2] - ((13*n - 91)*x + (10*n^2 - 151*n + 671)*x^2 - (n^2 - 123*n + 900)*x^3 - (12*n - 316)*x^4 - 32*x^5)*deriv(v[i - 2], x) - ((3*n - 21)*x^2 + (3*n^2 - 45*n + 192)*x^3 + (36*n - 276)*x^4 + 96*x^5)*deriv(deriv(v[i - 2], x), x) + (9*n - 71 + (2*n^2 - 31*n + 192)*x + (20*n - 156)*x^2 + 32*x^3)*v[i - 1] + ((13*n - 104)*x - (n - 112)*x^2 - 8*x^3)*deriv(v[i - 1], x) + ((3*n - 24)*x^2 + 24*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 8 + 8*x)); 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 Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.?
  - 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.