A225468 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 11\,x + 15\,x^{2} + 6\,x^{3} - 3\,n x - 6\,n x^{2} - 3\,n x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	3														5
3	4	15	12													31
4	8	63	117	63												251
5	16	267	864	1017	405											2569
6	32	1191	5919	11376	9639	3024										31181
7	64	5571	39726	110844	149931	98820	25515									430471
8	128	26895	266517	1016541	1954854	2031723	1093014	239355								6629027
9	256	132171	1806708	9098739	23370201	34069923	28647270	12989079	2465478							112579825
10	512	655479	12464955	81050166	267365853	512818209	599132295	421828560	164989467	27600669						2087906165
11	1024	3265107	87907194	726240870	2995056891	7259006565	11049067359	10737825912	6490449567	2229195249	333051669					41911067407
12	2048	16297887	634659585	6581883771	33284855988	99166882662	188847342837	237526503732	197230269114	104305669920	31903227729	4303119330				903800714603
13	4096	81427995	4686059808	60487912341	369869569785	1328162658822	3077964880992	4811189754942	5145594899193	3724691226948	1749166943382	481871988900	59202612693			20812969939897
14	8192	407004807	35289775143	564272655948	4130273497635	17616765438702	48697939968534	91999843171380	121682807315640	113032122745257	72444719163816	30570806040282	7656473255742	863285928327		509295005969405
15	16384	2034729123	270107737446	5345392438944	46502358580107	232993550807874	756740822302323	1693194022412892	2693095309895706	3082128070315293	2527567112922138	1452265998398370	556067692452312	127602924026427	13288589222508	13187063986257847

Technical details

Node id: n-a0ba4047ec7ad9c7 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-15d92847b79f4ccd

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 -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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
   1:   1;
   2:   2,      3;
   3:   4,     15,       12;
   4:   8,     63,      117,       63;
   5:  16,    267,      864,     1017,       405;
   6:  32,   1191,     5919,    11376,      9639,      3024;
   7:  64,   5571,    39726,   110844,    149931,     98820,     25515;
   8: 128,  26895,   266517,  1016541,   1954854,   2031723,   1093014,    239355;
   9: 256, 132171,  1806708,  9098739,  23370201,  34069923,  28647270,  12989079,   2465478;
  10: 512, 655479, 12464955, 81050166, 267365853, 512818209, 599132295, 421828560, 164989467, 27600669;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2-(3*n-11)*x-(6*n-15)*x^2-(3*n-6)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2 - (3*n - 11)*x - (6*n - 15)*x^2 - (3*n - 6)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2 - (3*n - 11)*x - (6*n - 15)*x^2 - (3*n - 6)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + x)); 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - 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

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.