A371081 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-54 + 21\,x + 2\,x^{2} - x^{3} - 81\,n + 31\,n x - 2\,n x^{2} - 45\,n^{2} + 14\,n^{2} x - n^{2} x^{2} - 11\,n^{3} + 2\,n^{3} x - n^{4}\right)\,P_{n-2}(x) + \left(-42\,x + 29\,x^{2} - 5\,x^{3} - 47\,n x + 22\,n x^{2} - 2\,n x^{3} - 17\,n^{2} x + 4\,n^{2} x^{2} - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-6\,x^{2} + 5\,x^{3} - x^{4} - 5\,n x^{2} + 2\,n x^{3} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(22 - 6\,x + x^{2} + 13\,n - 2\,n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(9\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 17 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	17	1							18
2	452	54	1						507
3	17348	3180	119	1					20648
4	908672	227760	12992	220	1				1149645
5	62312832	20029376	1552912	39920	365	1			83935406
6	5418432000	2142778688	212024320	7375872	102100	562	1		7780713543
7	582473664000	275245327872	33388763712	1476251840	27567412	229292	819	1	892611804948

All 15 rows

Technical details

Node id: n-abfce20d05838fa2 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f44590ad6843dfc1

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:                17,                1;
  2:               452,               54,                1;
  3:             17348,             3180,              119,              1;
  4:            908672,           227760,            12992,            220,             1;
  5:          62312832,         20029376,          1552912,          39920,           365,           1;
  6:        5418432000,       2142778688,        212024320,        7375872,        102100,         562,         1;
  7:      582473664000,     275245327872,      33388763712,     1476251840,      27567412,      229292,       819,      1;
  8:    75828980736000,   41898359098368,    6058231865088,   327172251136,    7740149952,    86651360,    467040,   1144,    1;
  9: 11752863796224000, 7467107308720128, 1260392761910016, 80877640516864, 2329814281920, 32962698304, 238725632, 881472, 1545, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4+11*n^3+45*n^2+81*n+54-(2*n^3+14*n^2+31*n+21)*x+(n^2+2*n-2)*x^2+x^3)*P(n-2,x) - ((2*n^3+17*n^2+47*n+42)*x-(4*n^2+22*n+29)*x^2+(2*n+5)*x^3)*d/dx P(n-2,x) - ((n^2+5*n+6)*x^2-(2*n+5)*x^3+x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+13*n+22-(2*n+6)*x+x^2)*P(n-1,x) + (2*n+9)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 17 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 17 + x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 11*n^3 + 45*n^2 + 81*n + 54 - (2*n^3 + 14*n^2 + 31*n + 21)*x + (n^2 + 2*n - 2)*x^2 + x^3)*P[n - 2, x] - ((2*n^3 + 17*n^2 + 47*n + 42)*x - (4*n^2 + 22*n + 29)*x^2 + (2*n + 5)*x^3)*D[P[n - 2, x], x] - ((n^2 + 5*n + 6)*x^2 - (2*n + 5)*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 13*n + 22 - (2*n + 6)*x + x^2)*P[n - 1, x] + (2*n + 9)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, 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; v[2] = 17 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 11*n^3 + 45*n^2 + 81*n + 54 - (2*n^3 + 14*n^2 + 31*n + 21)*x + (n^2 + 2*n - 2)*x^2 + x^3)*v[i - 2] - ((2*n^3 + 17*n^2 + 47*n + 42)*x - (4*n^2 + 22*n + 29)*x^2 + (2*n + 5)*x^3)*deriv(v[i - 2], x) - ((n^2 + 5*n + 6)*x^2 - (2*n + 5)*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 13*n + 22 - (2*n + 6)*x + x^2)*v[i - 1] + (2*n + 9)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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 read by rows, (2, 2)-Lah numbers.?
  - 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.