A177267: Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having genus k (see first comment for definition of genus).

A177267 ↗ nonnegative coefficients

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

Recurrence from A177267

\(\left(1 + \frac{1}{3}\,n\right)\,P_{n}(x) = \left(2 + \frac{4}{3}\,n\right)\,P_{n-1}(x) + \left(\frac{1}{3}\,n x + \frac{2}{3}\,n^{2} x + \frac{1}{3}\,n^{3} x\right)\,P_{n-2}(x)\)

\(P_{1}(x) = 5 + x\)

\(P_{2}(x) = 14 + 10\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	5	1								6
2	14	10								24
3	42	70	8							120
4	132	420	168							720
5	429	2310	2121	180						5040
6	1430	12012	20790	6088						40320
7	4862	60060	174174	115720	8064					362880
8	16796	291720	1309308	1624480	386496					3628800
9	58786	1385670	9087078	18748730	10031736	604800				39916800
10	208012	6466460	59306676	188208020	186698512	38113920				479001600
11	742900	29745716	368588220	1700309468	2788065280	1271140416	68428800			6227020800
12	2674440	135207800	2201182984	14139376680	35482759312	29817744320	5399345664			87178291200
13	9694845	608435100	12716293590	109952914500	399399285285	552088719000	222001739280	10897286400		1307674368000
14	35357670	2714556600	71430280740	808871005800	4076576013510	8588246612400	6334706026080	1040210035200		20922789888000
15	129644790	12021607800	391710517380	5678669653800	38404067142870	116695418439600	141006182148960	51174474508800	2324754432000	355687428096000

b-files: A124657 (row sums) ?

Technical details

Node id: n-3cde4de269220cd9 · revision 0 · JSON
Definition source: oeis_catalog:A177267
Fingerprint: f6-38707c17c5f06fdb

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 1)
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 ?

Row sums ? A000142, A109095, A354074, A342856, A052624, A338112, A124355, A124657

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      5,       1;
   2:     14,      10;
   3:     42,      70,        8;
   4:    132,     420,      168;
   5:    429,    2310,     2121,       180;
   6:   1430,   12012,    20790,      6088;
   7:   4862,   60060,   174174,    115720,      8064;
   8:  16796,  291720,  1309308,   1624480,    386496;
   9:  58786, 1385670,  9087078,  18748730,  10031736,   604800;
  10: 208012, 6466460, 59306676, 188208020, 186698512, 38113920;

%F Formula

The row polynomials satisfy ((1/3)*n+1)*P(n,x) = ((4/3)*n+2)*P(n-1,x) + ((1/3)*n^3+(2/3)*n^2+(1/3)*n)*x*P(n-2,x) for n >= 3, with P(1,x) = 5 + x, P(2,x) = 14 + 10*x.

%t Mathematica

P[1, x_] := 5 + x;
P[2, x_] := 14 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[(((4/3)*n + 2)*P[n - 1, x] + ((1/3)*n^3 + (2/3)*n^2 + (1/3)*n)*x*P[n - 2, x])/((1/3)*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 5 + x; v[2] = 14 + 10*x; for(n = 3, N, my(i = n); v[i] = (((4/3)*n + 2)*v[i - 1] + ((1/3)*n^3 + (2/3)*n^2 + (1/3)*n)*x*v[i - 2])/((1/3)*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); 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 ≥ 1, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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