A271703: Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.

A271703 ↗ nonnegative coefficients

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

Recurrence from A271703

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	0	1							1
3	0	2	1						3
4	0	6	6	1					13
5	0	24	36	12	1				73
6	0	120	240	120	20	1			501
7	0	720	1800	1200	300	30	1		4051
8	0	5040	15120	12600	4200	630	42	1	37633

All 15 rows b-files: A000262 (row sums), A089231 (reversed triangle), A105278, A271703 ?

Technical details

Node id: n-9e4edf560ed848c1 · revision 0 · JSON
Definition source: oeis_catalog:A271703
Fingerprint: f6-8fea465cea2922a0

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 ?

A105278 triangle 47 terms, 9 rows · tabl · nonn · catalog b-file?
A271703 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A089231 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A000262

Submission text ?

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

%e Example

Triangle begins:
   1: 1;
   2: 0,      1;
   3: 0,      2,       1;
   4: 0,      6,       6,       1;
   5: 0,     24,      36,      12,      1;
   6: 0,    120,     240,     120,     20,      1;
   7: 0,    720,    1800,    1200,    300,     30,     1;
   8: 0,   5040,   15120,   12600,   4200,    630,    42,    1;
   9: 0,  40320,  141120,  141120,  58800,  11760,  1176,   56,  1;
  10: 0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1;

%F Formula

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

A089231 reversed triangle, offset 1, from P₂, low zeros dropped

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,    6;
   4: 1, 12,   36,    24;
   5: 1, 20,  120,   240,    120;
   6: 1, 30,  300,  1200,   1800,     720;
   7: 1, 42,  630,  4200,  12600,   15120,     5040;
   8: 1, 56, 1176, 11760,  58800,  141120,   141120,    40320;
   9: 1, 72, 2016, 28224, 211680,  846720,  1693440,  1451520,   362880;
  10: 1, 90, 3240, 60480, 635040, 3810240, 12700800, 21772800, 16329600, 3628800;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + 2*x*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = x.

%t Mathematica

P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + 2*x*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + 2*x*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A105278 triangle, offset 1, from P₂, low zeros dropped

%e Example

Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       36,       12,        1;
   5:     120,      240,      120,       20,       1;
   6:     720,     1800,     1200,      300,      30,      1;
   7:    5040,    15120,    12600,     4200,     630,     42,     1;
   8:   40320,   141120,   141120,    58800,   11760,   1176,    56,    1;
   9:  362880,  1451520,  1693440,   846720,  211680,  28224,  2016,   72,  1;
  10: 3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + 2*x*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = x.

%t Mathematica

P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + 2*x*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + 2*x*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A271703 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0: 1;
  1: 0,      1;
  2: 0,      2,       1;
  3: 0,      6,       6,       1;
  4: 0,     24,      36,      12,      1;
  5: 0,    120,     240,     120,     20,      1;
  6: 0,    720,    1800,    1200,    300,     30,     1;
  7: 0,   5040,   15120,   12600,   4200,    630,    42,    1;
  8: 0,  40320,  141120,  141120,  58800,  11760,  1176,   56,  1;
  9: 0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1;

%F Formula

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

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.