A371259 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-144 - 18\,x - x^{2} - 168\,n - 12\,n x - 73\,n^{2} - 2\,n^{2} x - 14\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-81\,x - 8\,x^{2} - 72\,n x - 2\,n x^{2} - 21\,n^{2} x - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-9\,x^{2} - x^{3} - 6\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(34 + 2\,x + 16\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(11\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 36 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	36	2							38
2	1764	200	3						1967
3	112896	19728	600	4					133228
4	9144576	2198592	102192	1376	5				11446741
5	914457600	284318784	18013536	368800	2700	6			1217161426
6	110649369600	42770820096	3470765760	96302848	1068500	4776	7		156988331587
7	15933509222400	7449567335424	744426319104	26295828480	393914560	2662464	7840	8	24154195290280

All 15 rows

Technical details

Node id: n-e0b9881fafc1620d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-549c91eb45245c99

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:                 36,                  2;
  2:               1764,                200,                 3;
  3:             112896,              19728,               600,                4;
  4:            9144576,            2198592,            102192,             1376,              5;
  5:          914457600,          284318784,          18013536,           368800,           2700,            6;
  6:       110649369600,        42770820096,        3470765760,         96302848,        1068500,         4776,          7;
  7:     15933509222400,      7449567335424,      744426319104,      26295828480,      393914560,      2662464,       7840,        8;
  8:   2692763058585600,   1491982216187904,   178670272801536,    7724300516352,   146711093440,   1335335808,    5936448,    12160,     9;
  9: 527781559482777600, 341081524759449600, 47977563161475072, 2470549879627776, 57189769920000, 658109538816, 3932470976, 12147072, 18036, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4+14*n^3+73*n^2+168*n+144+(2*n^2+12*n+18)*x+x^2)*P(n-2,x) - ((2*n^3+21*n^2+72*n+81)*x+(2*n+8)*x^2)*d/dx P(n-2,x) - ((n^2+6*n+9)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+16*n+34+2*x)*P(n-1,x) + (2*n+11)*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) = 36 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 36 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (2*n^2 + 12*n + 18)*x + x^2)*P[n - 2, x] - ((2*n^3 + 21*n^2 + 72*n + 81)*x + (2*n + 8)*x^2)*D[P[n - 2, x], x] - ((n^2 + 6*n + 9)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 16*n + 34 + 2*x)*P[n - 1, x] + (2*n + 11)*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] = 36 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (2*n^2 + 12*n + 18)*x + x^2)*v[i - 2] - ((2*n^3 + 21*n^2 + 72*n + 81)*x + (2*n + 8)*x^2)*deriv(v[i - 2], x) - ((n^2 + 6*n + 9)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 16*n + 34 + 2*x)*v[i - 1] + (2*n + 11)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Triangle read by rows, (2, 3)-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.