A286724: Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].

A286724 ↗ nonnegative coefficients

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

Recurrence from A286724

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	8	8	1						17
4	48	72	18	1					139
5	384	768	288	32	1				1473
6	3840	9600	4800	800	50	1			19091
7	46080	138240	86400	19200	1800	72	1		291793
8	645120	2257920	1693440	470400	58800	3528	98	1	5129307

All 15 rows b-files: A025167 (row sums), A286724 ?

Technical details

Node id: n-24ab55cfc575f41e · revision 1 · JSON
Definition source: oeis_catalog:A286724
Fingerprint: f6-31dfc9d837f82561

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 ?

A286724 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A025167

Submission text ?

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

%e Example

Triangle begins:
   1:         1;
   2:         2,         1;
   3:         8,         8,         1;
   4:        48,        72,        18,         1;
   5:       384,       768,       288,        32,        1;
   6:      3840,      9600,      4800,       800,       50,       1;
   7:     46080,    138240,     86400,     19200,     1800,      72,      1;
   8:    645120,   2257920,   1693440,    470400,    58800,    3528,     98,     1;
   9:  10321920,  41287680,  36126720,  12042240,  1881600,  150528,   6272,   128,   1;
  10: 185794560, 836075520, 836075520, 325140480, 60963840, 6096384, 338688, 10368, 162, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (4+4*x)*d/dx P(n-1,x) + 4*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] + (4 + 4*x)*D[P[n - 1, x], x] + 4*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] + (4 + 4*x)*deriv(v[i - 1], x) + 4*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

A286724 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:         1;
  1:         2,         1;
  2:         8,         8,         1;
  3:        48,        72,        18,         1;
  4:       384,       768,       288,        32,        1;
  5:      3840,      9600,      4800,       800,       50,       1;
  6:     46080,    138240,     86400,     19200,     1800,      72,      1;
  7:    645120,   2257920,   1693440,    470400,    58800,    3528,     98,     1;
  8:  10321920,  41287680,  36126720,  12042240,  1881600,  150528,   6272,   128,   1;
  9: 185794560, 836075520, 836075520, 325140480, 60963840, 6096384, 338688, 10368, 162, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (4+4*x)*d/dx P(n-1,x) + 4*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] + (4 + 4*x)*D[P[n - 1, x], x] + 4*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] + (4 + 4*x)*deriv(v[i - 1], x) + 4*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 6

Related ?

Same fingerprint ?

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.