A292219: Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].

A292219 ↗ nonnegative coefficients

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

Recurrence from A292219

\(P_{n}(x) = \left(6 + x\right)\,P_{n-1}(x) + \left(24 + 8\,x\right)\,P_{n-1}'(x) + \left(16\,x\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	6	1							7
3	60	20	1						81
4	840	420	42	1					1303
5	15120	10080	1512	72	1				26785
6	332640	277200	55440	3960	110	1			669351
7	8648640	8648640	2162160	205920	8580	156	1		19674097
8	259459200	302702400	90810720	10810800	600600	16380	210	1	664400311

All 15 rows b-files: A292219 ?

Technical details

Node id: n-66d7d2fd3a2f1f0d · revision 1 · JSON
Definition source: oeis_catalog:A292219
Fingerprint: f6-f136838b00e1c323

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 -1)
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 ?

A292219 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:            1;
   2:            6,            1;
   3:           60,           20,            1;
   4:          840,          420,           42,           1;
   5:        15120,        10080,         1512,          72,          1;
   6:       332640,       277200,        55440,        3960,        110,         1;
   7:      8648640,      8648640,      2162160,      205920,       8580,       156,       1;
   8:    259459200,    302702400,     90810720,    10810800,     600600,     16380,     210,     1;
   9:   8821612800,  11762150400,   4116752640,   588107520,   40840800,   1485120,   28560,   272,   1;
  10: 335221286400, 502831929600, 201132771840, 33522128640, 2793510720, 126977760, 3255840, 46512, 342, 1;

%F Formula

The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + (24+8*x)*d/dx P(n-1,x) + 16*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(6 + x)*P[n - 1, x] + (24 + 8*x)*D[P[n - 1, x], x] + 16*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (6 + x)*v[i - 1] + (24 + 8*x)*deriv(v[i - 1], x) + 16*x*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, v[i])))};
T(10)

polytool c48629374597

A292219 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:            1;
  1:            6,            1;
  2:           60,           20,            1;
  3:          840,          420,           42,           1;
  4:        15120,        10080,         1512,          72,          1;
  5:       332640,       277200,        55440,        3960,        110,         1;
  6:      8648640,      8648640,      2162160,      205920,       8580,       156,       1;
  7:    259459200,    302702400,     90810720,    10810800,     600600,     16380,     210,     1;
  8:   8821612800,  11762150400,   4116752640,   588107520,   40840800,   1485120,   28560,   272,   1;
  9: 335221286400, 502831929600, 201132771840, 33522128640, 2793510720, 126977760, 3255840, 46512, 342, 1;

%F Formula

The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + (24+8*x)*d/dx P(n-1,x) + 16*x*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[(6 + x)*P[n - 1, x] + (24 + 8*x)*D[P[n - 1, x], x] + 16*x*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (6 + x)*v[i - 1] + (24 + 8*x)*deriv(v[i - 1], x) + 16*x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Iterated real-rootedness preservers
Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0)
by Row-hypergeometric rows (multiplier sequences)

Proof chains ? JSON

Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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.