A097805 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	2	12	16													30
3	4	32	80	64												180
4	8	80	288	448	256											1080
5	16	192	896	2048	2304	1024										6480
6	32	448	2560	7680	12800	11264	4096									38880
7	64	1024	6912	25600	56320	73728	53248	16384								233280
8	128	2304	17920	78848	215040	372736	401408	245760	65536							1399680
9	256	5120	45056	229376	745472	1605632	2293760	2097152	1114112	262144						8398080
10	512	11264	110592	638976	2408448	6193152	11010048	13369344	10616832	4980736	1048576					50388480
11	1024	24576	266240	1720320	7372800	22020096	46792704	70778880	74711040	52428800	22020096	4194304				302330880
12	2048	53248	630784	4505600	21626880	73531392	181665792	328728576	432537600	403701760	253755392	96468992	16777216			1813985280
13	4096	114688	1474560	11534336	61276160	233570304	657457152	1384120320	2179989504	2537553920	2122317824	1207959552	419430400	67108864		10883911680
14	8192	245760	3407872	28966912	168689664	712245248	2249195520	5398069248	9896460288	13795065856	14394851328	10905190400	5670699008	1811939328	268435456	65303470080

Technical details

Node id: n-f5f6d7397eb76ee7 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e0f7dc6dd641b7f6

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:   1,    4;
  2:   2,   12,    16;
  3:   4,   32,    80,     64;
  4:   8,   80,   288,    448,    256;
  5:  16,  192,   896,   2048,   2304,    1024;
  6:  32,  448,  2560,   7680,  12800,   11264,    4096;
  7:  64, 1024,  6912,  25600,  56320,   73728,   53248,   16384;
  8: 128, 2304, 17920,  78848, 215040,  372736,  401408,  245760,   65536;
  9: 256, 5120, 45056, 229376, 745472, 1605632, 2293760, 2097152, 1114112, 262144;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (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 Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read b…?
  - 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

Related ?

Same fingerprint ?

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.