A024462: Triangle T(n,k) read by rows, arising in enumeration of catafusenes.

A024462 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A024462

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

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

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

\(P_{2}(x) = 1 + 2\,x + x^{2}\)

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	2	1													4
3	1	5	7	3												16
4	1	8	22	24	9											64
5	1	11	46	90	81	27										256
6	1	14	79	228	351	270	81									1024
7	1	17	121	465	1035	1323	891	243								4096
8	1	20	172	828	2430	4428	4860	2916	729							16384
9	1	23	232	1344	4914	11718	18144	17496	9477	2187						65536
10	1	26	301	2040	8946	26460	53298	71928	61965	30618	6561					262144
11	1	29	379	2943	15066	53298	132678	231822	277749	216513	98415	19683				1048576
12	1	32	466	4080	23895	98496	292572	629856	973215	1049760	747954	314928	59049			4194304
13	1	35	562	5478	36135	170181	588060	1507572	2862783	3969405	3897234	2558790	1003833	177147		16777216
14	1	38	667	7164	52569	278586	1098603	3271752	7385499	12557754	15805449	14250492	8680203	3188646	531441	67108864

b-files: A024462, A155543 (row sums), A323943 (reversed triangle) ?

Technical details

Node id: n-df53153327eff8f9 · revision 0 · JSON
Definition source: oeis_catalog:A024462
Fingerprint: f6-fedd0649071087c4

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): yes

Matches ?

A024462 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
A323943 reversed triangle 64 terms, 9 rows · tabf · nonn b-file?

Row sums ? A155543, A000302

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  2,   1;
  3: 1,  5,   7,    3;
  4: 1,  8,  22,   24,    9;
  5: 1, 11,  46,   90,   81,    27;
  6: 1, 14,  79,  228,  351,   270,    81;
  7: 1, 17, 121,  465, 1035,  1323,   891,   243;
  8: 1, 20, 172,  828, 2430,  4428,  4860,  2916,  729;
  9: 1, 23, 232, 1344, 4914, 11718, 18144, 17496, 9477, 2187;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 2*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (3*n - 6)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (3*n - 6)*x)*v[i - 1])/(n - 2)); 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

A024462 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  2,   1;
  3: 1,  5,   7,    3;
  4: 1,  8,  22,   24,    9;
  5: 1, 11,  46,   90,   81,    27;
  6: 1, 14,  79,  228,  351,   270,    81;
  7: 1, 17, 121,  465, 1035,  1323,   891,   243;
  8: 1, 20, 172,  828, 2430,  4428,  4860,  2916,  729;
  9: 1, 23, 232, 1344, 4914, 11718, 18144, 17496, 9477, 2187;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 2*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (3*n - 6)*x)*P[n - 1, x])/(n - 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; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (3*n - 6)*x)*v[i - 1])/(n - 2)); 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

A323943 reversed triangle, offset 1, from P₂

%e Example

Triangle begins:
   1:     1,     2,      1;
   2:     3,     7,      5,      1;
   3:     9,    24,     22,      8,      1;
   4:    27,    81,     90,     46,     11,      1;
   5:    81,   270,    351,    228,     79,     14,     1;
   6:   243,   891,   1323,   1035,    465,    121,    17,     1;
   7:   729,  2916,   4860,   4428,   2430,    828,   172,    20,    1;
   8:  2187,  9477,  17496,  18144,  11718,   4914,  1344,   232,   23,   1;
   9:  6561, 30618,  61965,  71928,  53298,  26460,  8946,  2040,  301,  26,  1;
  10: 19683, 98415, 216513, 277749, 231822, 132678, 53298, 15066, 2943, 379, 29, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (n-1+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 2*x + x^2.

%t Mathematica

P[1, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (3*n - 3)*x)*P[n - 1, x])/(n - 1)]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 2*x + x^2; for(n = 2, N, my(i = n); v[i] = ((n - 1 + (3*n - 3)*x)*v[i - 1])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(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 Liu–Wang theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem, Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem, Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 3
by Liu–Wang theorem
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by 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

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Used in 5

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.