A371259 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 16\,x + 16\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-7\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	36							37
2	1	100	1764						1865
3	1	200	9864	112896					122961
4	1	344	34064	1099296	9144576				10278281
5	1	540	92200	6004512	142159392	914457600			1062714245
6	1	796	213700	24075712	1156921920	21385410048	110649369600		133215991777
7	1	1120	443744	78782912	6573957120	248142106368	3724783667712	15933509222400	19913088181377

All 15 rows

Technical details

Node id: n-1e28f4cefba7f739 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-4018c0defb865ce3

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: 1,   36;
  2: 1,  100,    1764;
  3: 1,  200,    9864,    112896;
  4: 1,  344,   34064,   1099296,      9144576;
  5: 1,  540,   92200,   6004512,    142159392,      914457600;
  6: 1,  796,  213700,  24075712,   1156921920,    21385410048,    110649369600;
  7: 1, 1120,  443744,  78782912,   6573957120,   248142106368,   3724783667712,    15933509222400;
  8: 1, 1520,  848064, 222555968,  29342218688,  1931075129088,  59556757600512,   745991108093952,   2692763058585600;
  9: 1, 2004, 1518384, 561781568, 109684923136, 11437953984000, 617637469906944, 15992521053825024, 170540762379724800, 527781559482777600;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n^2+16*n+16)*x)*P(n-1,x) - (4*n+7)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 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[(1 + (4*n^2 + 16*n + 16)*x)*P[n - 1, x] - (4*n + 7)*x^2*D[P[n - 1, x], x] + x^3*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; for(n = 1, N, my(i = n + 1); v[i] = (1 + (4*n^2 + 16*n + 16)*x)*v[i - 1] - (4*n + 7)*x^2*deriv(v[i - 1], x) + x^3*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) 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

Related ?

Same fingerprint ?

A371259: Triangle read by rows, (2, 3)-Lah numbers.

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.