A156319 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4														7
2	3	16	16													35
3	3	28	80	64												175
4	3	40	192	384	256											875
5	3	52	352	1152	1792	1024										4375
6	3	64	560	2560	6400	8192	4096									21875
7	3	76	816	4800	16640	33792	36864	16384								109375
8	3	88	1120	8064	35840	100352	172032	163840	65536							546875
9	3	100	1472	12544	68096	243712	573440	851968	720896	262144						2734375
10	3	112	1872	18432	118272	516096	1548288	3145728	4128768	3145728	1048576					13671875
11	3	124	2320	25920	192000	989184	3612672	9338880	16711680	19660800	13631488	4194304				68359375
12	3	136	2816	35200	295680	1757184	7569408	23789568	54067200	86507520	92274688	58720256	16777216			341796875
13	3	148	3360	46464	436480	2939904	14598144	54067200	149225472	302776320	438304768	427819008	251658240	67108864		1708984375
14	3	160	3952	59904	622336	4685824	26357760	112459776	365494272	899678208	1649410048	2181038080	1962934272	1073741824	268435456	8544921875

Technical details

Node id: n-ed4b2207be4674b7 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-d00b34d97cb36be9

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 3,   4;
  2: 3,  16,   16;
  3: 3,  28,   80,    64;
  4: 3,  40,  192,   384,   256;
  5: 3,  52,  352,  1152,  1792,   1024;
  6: 3,  64,  560,  2560,  6400,   8192,   4096;
  7: 3,  76,  816,  4800, 16640,  33792,  36864,  16384;
  8: 3,  88, 1120,  8064, 35840, 100352, 172032, 163840,  65536;
  9: 3, 100, 1472, 12544, 68096, 243712, 573440, 851968, 720896, 262144;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (4*n - 4)*x)*P[n - 1, x])/(n - 1)]];
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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (4*n - 4)*x)*v[i - 1])/(n - 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle by columns: (1, 2, 0, 0, 0, ...) in every column.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - 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.