A120986 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-8\,n - 22\,n x - 20\,n x^{2} - 6\,n x^{3} + 14\,n^{2} + 46\,n^{2} x + 50\,n^{2} x^{2} + 18\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 6\,x - 12\,x^{2} - 10\,x^{3} - 3\,x^{4} + n + 12\,n x + 30\,n x^{2} + 28\,n x^{3} + 9\,n x^{4}\right)\,P_{n-1}'(x)\)

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

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	5	16	12													33
4	14	70	110	55												249
5	42	288	702	728	273											2033
6	132	1155	3850	6160	4760	1428										17485
7	429	4576	19448	42432	50388	31008	7752									156033
8	1430	18018	93366	259350	418950	395010	201894	43263								1431281
9	4862	70720	433160	1466080	3010700	3853696	3010700	1315600	246675							13412193
10	16796	277134	1961256	7845024	19612560	31870410	33745140	22496760	8584290	1430715						127840085
11	58786	1085280	8721000	40310400	119041650	235282320	315888300	285012000	165663225	56097600	8414640					1235575201
12	208012	4249388	38244492	200783583	685026342	1598394798	2607907302	2980465488	2341794312	1206378888	367158792	50067108				12080678505
13	742900	16640960	165889570	975821000	3781306375	10189625600	19615029280	27220856960	27066193000	18828656000	8708253400	2406644576	300830572			119276490193
14	2674440	65189475	713249550	4649330400	20187882000	61774918920	137277597600	224636068800	271028300400	238404523500	148764422664	62418638880	15797309840	1822766520		1187542872989
15	9694845	255487680	3044561520	21792650880	104877132360	359578739520	907119092880	1712821144320	2435417564580	2597778735552	2048248618416	1158605279104	444821669656	103831052864	11124755664	11909326179841

Technical details

Node id: n-8f2b5ae45609013a · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-031025df87457cd9

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:     5,     16,      12;
   4:    14,     70,     110,      55;
   5:    42,    288,     702,     728,      273;
   6:   132,   1155,    3850,    6160,     4760,     1428;
   7:   429,   4576,   19448,   42432,    50388,    31008,     7752;
   8:  1430,  18018,   93366,  259350,   418950,   395010,   201894,    43263;
   9:  4862,  70720,  433160, 1466080,  3010700,  3853696,  3010700,  1315600,  246675;
  10: 16796, 277134, 1961256, 7845024, 19612560, 31870410, 33745140, 22496760, 8584290, 1430715;

%F Formula

The row polynomials satisfy (4*n^2+2*n+(8*n^2+4*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (14*n^2-8*n+(46*n^2-22*n)*x+(50*n^2-20*n)*x^2+(18*n^2-6*n)*x^3)*P(n-1,x) + (n-1+(12*n-6)*x+(30*n-12)*x^2+(28*n-10)*x^3+(9*n-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[((14*n^2 - 8*n + (46*n^2 - 22*n)*x + (50*n^2 - 20*n)*x^2 + (18*n^2 - 6*n)*x^3)*P[n - 1, x] + (n - 1 + (12*n - 6)*x + (30*n - 12)*x^2 + (28*n - 10)*x^3 + (9*n - 3)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (8*n^2 + 4*n)*x + (4*n^2 + 2*n)*x^2)]];
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] = ((14*n^2 - 8*n + (46*n^2 - 22*n)*x + (50*n^2 - 20*n)*x^2 + (18*n^2 - 6*n)*x^3)*v[i - 1] + (n - 1 + (12*n - 6)*x + (30*n - 12)*x^2 + (28*n - 10)*x^3 + (9*n - 3)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (8*n^2 + 4*n)*x + (4*n^2 + 2*n)*x^2)); 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: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
  - 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

References

[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.