A048854: Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].

A048854 ↗ nonnegative coefficients

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

Recurrence from A048854

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(8 + 8\,x\right)\,P_{n-1}'(x) + \left(16\,x\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	12	12	1						25
4	120	180	30	1					331
5	1680	3360	840	56	1				5937
6	30240	75600	25200	2520	90	1			133651
7	665280	1995840	831600	110880	5940	132	1		3609673
8	17297280	60540480	30270240	5045040	360360	12012	182	1	113525595

All 15 rows b-files: A048854 ?

Technical details

Node id: n-be41df8499a2681b · revision 1 · JSON
Definition source: oeis_catalog:A048854
Fingerprint: f6-14fed6d19df1fb72

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 ?

A048854 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Submission text ?

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

%e Example

Triangle begins:
   1:           1;
   2:           2,           1;
   3:          12,          12,           1;
   4:         120,         180,          30,           1;
   5:        1680,        3360,         840,          56,          1;
   6:       30240,       75600,       25200,        2520,         90,        1;
   7:      665280,     1995840,      831600,      110880,       5940,      132,       1;
   8:    17297280,    60540480,    30270240,     5045040,     360360,    12012,     182,     1;
   9:   518918400,  2075673600,  1210809600,   242161920,   21621600,   960960,   21840,   240,   1;
  10: 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (8+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[(2 + x)*P[n - 1, x] + (8 + 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] = (2 + x)*v[i - 1] + (8 + 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

A048854 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:           1;
  1:           2,           1;
  2:          12,          12,           1;
  3:         120,         180,          30,           1;
  4:        1680,        3360,         840,          56,          1;
  5:       30240,       75600,       25200,        2520,         90,        1;
  6:      665280,     1995840,      831600,      110880,       5940,      132,       1;
  7:    17297280,    60540480,    30270240,     5045040,     360360,    12012,     182,     1;
  8:   518918400,  2075673600,  1210809600,   242161920,   21621600,   960960,   21840,   240,   1;
  9: 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (8+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[(2 + x)*P[n - 1, x] + (8 + 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] = (2 + x)*v[i - 1] + (8 + 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 Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
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

Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
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

Used in 1

Coefficient array of orthogonal polynomials whose moment sequence is the double… Image under a real-rootedness preserver (affine, shift 2)

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.