A241171: Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.

A241171 ↗ nonnegative coefficients

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

Recurrence from A241171

\(P_{n}(x) = \left(1 + 6\,x\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	6							7
2	1	30	90						121
3	1	126	1260	2520					3907
4	1	510	13230	75600	113400				202741
5	1	2046	126720	1580040	6237000	7484400			15430207
6	1	8190	1171170	28828800	227026800	681080400	681080400		1619195761
7	1	32766	10663380	494053560	6972966000	39502663200	95351256000	81729648000	224061282907

All 15 rows b-files: A241171 ?

Technical details

Node id: n-4ef34abfaa61f0d6 · revision 0 · JSON
Definition source: oeis_catalog:A241171
Fingerprint: f6-18f46c82cbc91532

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 ?

A241171 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A094088

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,      6;
  2: 1,     30,        90;
  3: 1,    126,      1260,         2520;
  4: 1,    510,     13230,        75600,        113400;
  5: 1,   2046,    126720,      1580040,       6237000,        7484400;
  6: 1,   8190,   1171170,     28828800,     227026800,      681080400,       681080400;
  7: 1,  32766,  10663380,    494053560,    6972966000,    39502663200,     95351256000,      81729648000;
  8: 1, 131070,  96461910,   8203431600,  196556560200,  1882311631200,   8266953895200,   16672848192000,   12504636144000;
  9: 1, 524286, 870123240, 133955839080, 5283068427000, 80735951696400, 576371099304000, 2059096751712000, 3563821301040000, 2375880867360000;

%F Formula

The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) + (3*x+9*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*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 + 6*x)*P[n - 1, x] + (3*x + 9*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*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] = (1 + 6*x)*v[i - 1] + (3*x + 9*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*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

A241171 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,      6;
   3: 1,     30,        90;
   4: 1,    126,      1260,         2520;
   5: 1,    510,     13230,        75600,        113400;
   6: 1,   2046,    126720,      1580040,       6237000,        7484400;
   7: 1,   8190,   1171170,     28828800,     227026800,      681080400,       681080400;
   8: 1,  32766,  10663380,    494053560,    6972966000,    39502663200,     95351256000,      81729648000;
   9: 1, 131070,  96461910,   8203431600,  196556560200,  1882311631200,   8266953895200,   16672848192000,   12504636144000;
  10: 1, 524286, 870123240, 133955839080, 5283068427000, 80735951696400, 576371099304000, 2059096751712000, 3563821301040000, 2375880867360000;

%F Formula

The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) + (3*x+9*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*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 + 6*x)*P[n - 1, x] + (3*x + 9*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n], {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 + 6*x)*v[i - 1] + (3*x + 9*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); 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

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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by 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

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.