A166960: Triangle T(n, k) read by rows: T(n, k)= (mn-mk+1)T(n-1, k-1) + k(mk-(m-1))T(n-1, k) where m = 1.

A166960 ↗ nonnegative coefficients

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

Recurrence from A166960

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	1	6	1													8
3	1	27	21	1												50
4	1	112	270	58	1											442
5	1	453	2878	1738	141	1										5212
6	1	1818	28167	39320	8739	318	1									78364
7	1	7279	264411	769955	375755	37665	685	1								1455752
8	1	29124	2430652	13905746	13243650	2858960	146560	1434	1							32616128
9	1	116505	22108860	239506500	414525726	169140810	18617280	531456	2949	1						864550088
10	1	466030	200028285	4008974880	12039688650	8576223516	1757950770	108482304	1833237	5998	1					26693653672
11	1	1864131	1804914865	65943852645	333064015290	393021867126	137596928826	15732621306	582421413	6099511	12117	1				947754597232
12	1	7456536	16264739226	1073150790970	8920095056055	16813299338856	9493402582356	1832469336540	125839240983	2939636752	19764690	24378	1			38277487967344
13	1	29826157	146472131466	17349324787006	233733884311075	685559631703311	599683121246292	183731855615052	21187794538863	923159880115	14150074498	62804502	48925	1		1742329486967264
14	1	119304642	1318636923235	279346862169688	6034189680433941	27017485584429946	35554509626398107	16556303729333664	3002334346953267	219442755244678	6327958414833	65644146280	196681831	98046	1	88671325140532160

b-files: A166960 ?

Technical details

Node id: n-0c97d42b0cd05e48 · revision 0 · JSON
Definition source: oeis_catalog:A166960
Fingerprint: f6-2b1205fb0a5fe514

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 ?

A166960 triangle 48 terms, 9 rows · tabl · nonn · catalog b-file?

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,      1;
  2: 1,      6,        1;
  3: 1,     27,       21,         1;
  4: 1,    112,      270,        58,         1;
  5: 1,    453,     2878,      1738,       141,         1;
  6: 1,   1818,    28167,     39320,      8739,       318,        1;
  7: 1,   7279,   264411,    769955,    375755,     37665,      685,      1;
  8: 1,  29124,  2430652,  13905746,  13243650,   2858960,   146560,   1434,    1;
  9: 1, 116505, 22108860, 239506500, 414525726, 169140810, 18617280, 531456, 2949, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) + (3*x-x^2)*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[(1 + n*x)*P[n - 1, x] + (3*x - x^2)*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] = (1 + n*x)*v[i - 1] + (3*x - x^2)*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

A166960 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,      6,        1;
   4: 1,     27,       21,         1;
   5: 1,    112,      270,        58,         1;
   6: 1,    453,     2878,      1738,       141,         1;
   7: 1,   1818,    28167,     39320,      8739,       318,        1;
   8: 1,   7279,   264411,    769955,    375755,     37665,      685,      1;
   9: 1,  29124,  2430652,  13905746,  13243650,   2858960,   146560,   1434,    1;
  10: 1, 116505, 22108860, 239506500, 414525726, 169140810, 18617280, 531456, 2949, 1;

%F Formula

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

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 ?

A166960 → 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.