A166972: Triangle T(n,k) read by rows: T(n,k) = (n-k+1)T(n-1,k-1) + (3k-2)kT(n-1,k), initialized by T(n,1) = T(n,n) = 1.

A166972 ↗ nonnegative coefficients

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

Recurrence from A166972

\(P_{n}(x) = \left(1 + n x\right)\,P_{n-1}(x) + \left(7\,x - x^{2}\right)\,P_{n-1}'(x) + \left(3\,x^{2}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	10	1						12
3	1	83	41	1					126
4	1	668	1110	122	1				1902
5	1	5349	25982	8210	309	1			39852
6	1	42798	572367	432328	44715	714	1		1092924
7	1	342391	12276495	20154955	4635787	202689	1561	1	37613880

All 15 rows b-files: A166972 ?

Technical details

Node id: n-56eee146d6109546 · revision 0 · JSON
Definition source: oeis_catalog:A166972
Fingerprint: f6-e7b08fefd490fef1

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 ?

A166972 triangle 43 terms, 8 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,       10,          1;
  3: 1,       83,         41,           1;
  4: 1,      668,       1110,         122,           1;
  5: 1,     5349,      25982,        8210,         309,          1;
  6: 1,    42798,     572367,      432328,       44715,        714,         1;
  7: 1,   342391,   12276495,    20154955,     4635787,     202689,      1561,       1;
  8: 1,  2739136,  260203132,   879857170,   402100930,   38001292,    815680,    3298,    1;
  9: 1, 21913097, 5486178860, 37015708724, 31415703470, 5658628682, 260490608, 3027488, 6821, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) + (7*x-x^2)*d/dx P(n-1,x) + 3*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] + (7*x - x^2)*D[P[n - 1, x], x] + 3*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] + (7*x - x^2)*deriv(v[i - 1], x) + 3*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

A166972 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       10,          1;
   4: 1,       83,         41,           1;
   5: 1,      668,       1110,         122,           1;
   6: 1,     5349,      25982,        8210,         309,          1;
   7: 1,    42798,     572367,      432328,       44715,        714,         1;
   8: 1,   342391,   12276495,    20154955,     4635787,     202689,      1561,       1;
   9: 1,  2739136,  260203132,   879857170,   402100930,   38001292,    815680,    3298,    1;
  10: 1, 21913097, 5486178860, 37015708724, 31415703470, 5658628682, 260490608, 3027488, 6821, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (7*x-x^2)*d/dx P(n-1,x) + 3*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] + (7*x - x^2)*D[P[n - 1, x], x] + 3*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] + (7*x - x^2)*deriv(v[i - 1], x) + 3*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

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.