A093644 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 10 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	10	2														12
2	20	24	4													48
3	40	88	56	8												192
4	80	256	288	128	16											768
5	160	672	1088	832	288	32										3072
6	320	1664	3520	3840	2240	640	64									12288
7	640	3968	10368	14720	12160	5760	1408	128								49152
8	1280	9216	28672	50176	53760	35840	14336	3072	256							196608
9	2560	20992	75776	157696	207872	179200	100352	34816	6656	512						786432
10	5120	47104	193536	466944	731136	774144	559104	270336	82944	14336	1024					3145728
11	10240	104448	481280	1320960	2396160	3010560	2666496	1658880	706560	194560	30720	2048				12582912
12	20480	229376	1171456	3604480	7434240	10813440	11354112	8650752	4730880	1802240	450560	65536	4096			50331648
13	40960	499712	2801664	9551872	22077440	36495360	44335104	40009728	26763264	13066240	4505600	1032192	139264	8192		201326592
14	81920	1081344	6602752	24707072	63258624	117145600	161660928	168689664	133545984	79659008	35143680	11075584	2342912	294912	16384	805306368

Technical details

Node id: n-3de2c5407c0d896b · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-5bcf47c8067a6574

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:   10,     2;
  2:   20,    24,     4;
  3:   40,    88,    56,      8;
  4:   80,   256,   288,    128,     16;
  5:  160,   672,  1088,    832,    288,     32;
  6:  320,  1664,  3520,   3840,   2240,    640,     64;
  7:  640,  3968, 10368,  14720,  12160,   5760,   1408,   128;
  8: 1280,  9216, 28672,  50176,  53760,  35840,  14336,  3072,  256;
  9: 2560, 20992, 75776, 157696, 207872, 179200, 100352, 34816, 6656, 512;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(2*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 10 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (2*n - 2)*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] = 10 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (2*n - 2)*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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from (9,1) Pascal triangle.?
  - 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
  - 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

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
[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.