A371259: Triangle read by rows, (2, 3)-Lah numbers.

A371259 ↗ nonnegative coefficients

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

Recurrence from A371259

\(P_{n}(x) = \left(25 + x + 10\,n + n^{2}\right)\,P_{n-1}(x) + \left(11\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	36	1							37
2	1764	100	1						1865
3	112896	9864	200	1					122961
4	9144576	1099296	34064	344	1				10278281
5	914457600	142159392	6004512	92200	540	1			1062714245
6	110649369600	21385410048	1156921920	24075712	213700	796	1		133215991777
7	15933509222400	3724783667712	248142106368	6573957120	78782912	443744	1120	1	19913088181377

All 15 rows b-files: A371259, A371489 (row sums) ?

Technical details

Node id: n-b8808e9caf8dcf35 · revision 0 · JSON
Definition source: oeis_catalog:A371259
Fingerprint: f6-4018c0defb865ce3

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 0)
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 ?

A371259 triangle 36 terms, 8 rows · tabl · nonn · catalog b-file?

Row sums ? A371489

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:                  1;
  1:                 36,                  1;
  2:               1764,                100,                 1;
  3:             112896,               9864,               200,               1;
  4:            9144576,            1099296,             34064,             344,              1;
  5:          914457600,          142159392,           6004512,           92200,            540,            1;
  6:       110649369600,        21385410048,        1156921920,        24075712,         213700,          796,         1;
  7:     15933509222400,      3724783667712,      248142106368,      6573957120,       78782912,       443744,      1120,       1;
  8:   2692763058585600,    745991108093952,    59556757600512,   1931075129088,    29342218688,    222555968,    848064,    1520,    1;
  9: 527781559482777600, 170540762379724800, 15992521053825024, 617637469906944, 11437953984000, 109684923136, 561781568, 1518384, 2004, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n^2+10*n+25+x)*P(n-1,x) + (2*n+11)*x*d/dx P(n-1,x) + x^2*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[(n^2 + 10*n + 25 + x)*P[n - 1, x] + (2*n + 11)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {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] = (n^2 + 10*n + 25 + x)*v[i - 1] + (2*n + 11)*x*deriv(v[i - 1], x) + x^2*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 - 1, v[i])))};
T(9)

polytool c48629374597

A371259 triangle, offset 3, from P₀

%e Example

Triangle begins:
   3:                  1;
   4:                 36,                  1;
   5:               1764,                100,                 1;
   6:             112896,               9864,               200,               1;
   7:            9144576,            1099296,             34064,             344,              1;
   8:          914457600,          142159392,           6004512,           92200,            540,            1;
   9:       110649369600,        21385410048,        1156921920,        24075712,         213700,          796,         1;
  10:     15933509222400,      3724783667712,      248142106368,      6573957120,       78782912,       443744,      1120,       1;
  11:   2692763058585600,    745991108093952,    59556757600512,   1931075129088,    29342218688,    222555968,    848064,    1520,    1;
  12: 527781559482777600, 170540762379724800, 15992521053825024, 617637469906944, 11437953984000, 109684923136, 561781568, 1518384, 2004, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n^2+4*n+4+x)*P(n-1,x) + (2*n+5)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 4, with P(3,x) = 1.

%t Mathematica

P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 + 4*n + 4 + x)*P[n - 1, x] + (2*n + 5)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n - 2], {n, 3, 12}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N - 2)); v[1] = 1; for(n = 4, N, my(i = n - 2); v[i] = (n^2 + 4*n + 4 + x)*v[i - 1] + (2*n + 5)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};
T(12)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 7

Related ?

Same fingerprint ?

A371259 → reverse

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.