A059110: Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

A059110 ↗ nonnegative coefficients

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

Recurrence from A059110

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	1							2
3	3	4	1						8
4	13	21	9	1					44
5	73	136	78	16	1				304
6	501	1045	730	210	25	1			2512
7	4051	9276	7515	2720	465	36	1		24064
8	37633	93289	85071	36575	8015	903	49	1	261536

All 15 rows b-files: A052897 (row sums), A059110 ?

Technical details

Node id: n-01897e65d9bb3867 · revision 0 · JSON
Definition source: oeis_catalog:A059110
Fingerprint: f6-65341f67bf7e02f3

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 ?

A059110 triangle 47 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A052897

Submission text ?

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

%e Example

Triangle begins:
   1:       1;
   2:       1,        1;
   3:       3,        4,        1;
   4:      13,       21,        9,       1;
   5:      73,      136,       78,      16,       1;
   6:     501,     1045,      730,     210,      25,      1;
   7:    4051,     9276,     7515,    2720,     465,     36,     1;
   8:   37633,    93289,    85071,   36575,    8015,    903,    49,    1;
   9:  394353,  1047376,  1053724,  519456,  137270,  20048,  1596,   64,  1;
  10: 4596553, 12975561, 14196708, 7836276, 2404206, 427518, 44436, 2628, 81, 1;

%F Formula

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

A059110 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:       1;
  1:       1,        1;
  2:       3,        4,        1;
  3:      13,       21,        9,       1;
  4:      73,      136,       78,      16,       1;
  5:     501,     1045,      730,     210,      25,      1;
  6:    4051,     9276,     7515,    2720,     465,     36,     1;
  7:   37633,    93289,    85071,   36575,    8015,    903,    49,    1;
  8:  394353,  1047376,  1053724,  519456,  137270,  20048,  1596,   64,  1;
  9: 4596553, 12975561, 14196708, 7836276, 2404206, 427518, 44436, 2628, 81, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) + (1+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[(1 + x)*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x] + (1 + 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] = (1 + x)*v[i - 1] + (2 + 2*x)*deriv(v[i - 1], x) + (1 + 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
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers

Proof chains ? JSON

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

Used in 1

The matrix inverse of the unsigned Lah numbers A271703. Image under a real-rootedness preserver (affine, shift 0)

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.