basis-map coefficients of A371080, shifted → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 30\,x - 9\,x^{2} + 30\,n - 18\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 6\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	19	33	9													61
3	145	333	189	27												694
4	1534	4116	3213	918	81											9862
5	20782	61872	58410	23355	4050	243										168712
6	343432	1103724	1187271	584955	143370	16767	729									3380248
7	6699928	22856628	27097119	15453963	4723110	785862	66339	2187								77685136
8	150718096	539725704	690601392	439744032	157244409	32897340	3970134	253692	6561							2015161360
9	3840985360	14324418216	19504239804	13549330404	5459168988	1346748039	205630488	18856314	944784	19683						58250342080
10	109373414080	422303798208	605730030444	451852190100	200177165835	56027464857	10147769667	1183013910	85325805	3444525	59049					1856883676480
11	3441698908480	13696535156928	20539499450580	16265352310812	7789751772165	2411748595377	498282705459	69234434181	6373060155	371319795	12341241	177147				64718860232320
12	118602578120320	484747573554432	755607248779176	629788422373704	322059645055566	108360153048708	24884157821193	3961325745234	436286902293	32546234160	1564739451	43578162	531441			2448481546483840
13	4442179108339840	18593391972347904	29987870277587976	26132547505578696	14137024918103802	5102449828216884	1278816836679954	227168659653507	28768698072972	2576044176849	159000651810	6418212957	151992126	1594323		99932959421209600
14	179680861895449600	768395055982099200	1277518655097502848	1157845491974519712	657833054640420204	252200399753942640	68062692554131275	13242308454431199	1877553204468084	193904171474013	14405340837888	748259539209	25728121692	524532267	4782969	4376865157586252800

Technical details

Node id: n-7794429e5748ba6a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0a9a0c55c6b499ab

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:          1;
  1:          4,           3;
  2:         19,          33,           9;
  3:        145,         333,         189,          27;
  4:       1534,        4116,        3213,         918,         81;
  5:      20782,       61872,       58410,       23355,       4050,        243;
  6:     343432,     1103724,     1187271,      584955,     143370,      16767,       729;
  7:    6699928,    22856628,    27097119,    15453963,    4723110,     785862,     66339,     2187;
  8:  150718096,   539725704,   690601392,   439744032,  157244409,   32897340,   3970134,   253692,   6561;
  9: 3840985360, 14324418216, 19504239804, 13549330404, 5459168988, 1346748039, 205630488, 18856314, 944784, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(18*n-30)*x+9*x^2)*P(n-2,x) + (6*n-7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*P[n - 2, x] + (6*n - 7 + 6*x)*P[n - 1, x]];
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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*v[i - 2] + (6*n - 7 + 6*x)*v[i - 1]); 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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    ↑ Parent-row provers of polytool (above)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.