A290596: Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].

A290596 ↗ nonnegative coefficients

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

Recurrence from A290596

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(6 + 6\,x\right)\,P_{n-1}'(x) + \left(9\,x\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	10	10	1						21
4	80	120	24	1					225
5	880	1760	528	44	1				3213
6	12320	30800	12320	1540	70	1			57051
7	209440	628320	314160	52360	3570	102	1		1207953
8	4188800	14660800	8796480	1832600	166600	7140	140	1	29652561

All 15 rows b-files: A290596 ?

Technical details

Node id: n-33e7d067844832dd · revision 1 · JSON
Definition source: oeis_catalog:A290596
Fingerprint: f6-94f770baffeadbcb

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 ?

A290596 triangle 66 terms, 11 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:         10,          10,          1;
   4:         80,         120,         24,          1;
   5:        880,        1760,        528,         44,         1;
   6:      12320,       30800,      12320,       1540,        70,        1;
   7:     209440,      628320,     314160,      52360,      3570,      102,       1;
   8:    4188800,    14660800,    8796480,    1832600,    166600,     7140,     140,     1;
   9:   96342400,   385369600,  269758720,   67439680,   7663600,   437920,   12880,   184,   1;
  10: 2504902400, 11272060800, 9017648640, 2630147520, 358656480, 25618320, 1004640, 21528, 234, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (6+6*x)*d/dx P(n-1,x) + 9*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] + (6 + 6*x)*D[P[n - 1, x], x] + 9*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] + (6 + 6*x)*deriv(v[i - 1], x) + 9*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

A290596 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:          1;
  1:          2,           1;
  2:         10,          10,          1;
  3:         80,         120,         24,          1;
  4:        880,        1760,        528,         44,         1;
  5:      12320,       30800,      12320,       1540,        70,        1;
  6:     209440,      628320,     314160,      52360,      3570,      102,       1;
  7:    4188800,    14660800,    8796480,    1832600,    166600,     7140,     140,     1;
  8:   96342400,   385369600,  269758720,   67439680,   7663600,   437920,   12880,   184,   1;
  9: 2504902400, 11272060800, 9017648640, 2630147520, 358656480, 25618320, 1004640, 21528, 234, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (6+6*x)*d/dx P(n-1,x) + 9*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] + (6 + 6*x)*D[P[n - 1, x], x] + 9*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] + (6 + 6*x)*deriv(v[i - 1], x) + 9*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

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.