A088729 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 7\,x + 4\,x^{2} - 4\,n - 17\,n x - 24\,n x^{2} - 13\,n x^{3} - 2\,n x^{4} + 2\,n^{2} x + 6\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(4 + 23\,x + 49\,x^{2} + 49\,x^{3} + 23\,x^{4} + 4\,x^{5} - 4\,n x - 16\,n x^{2} - 24\,n x^{3} - 16\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x) + \left(2\,x + 10\,x^{2} + 20\,x^{3} + 20\,x^{4} + 10\,x^{5} + 2\,x^{6}\right)\,P_{n-1}''(x)\)

\(P_{1}(x) = 13 + 35\,x + 23\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	13	35	23															71
2	75	304	401	173														953
3	541	2929	5806	5019	1602													15897
4	4683	31726	83944	108971	69644	17575												316543
5	47293	384747	1273441	2205804	2116241	1068920	222497											7318943
6	545835	5183284	20597667	44622201	57112019	43295414	18034183	3188806										192579409
7	7087261	76944533	356868968	928109741	1485477001	1502110525	938928838	332185673	50988405									5678700945
8	102247563	1249200538	6623511626	20103093515	38623441709	48835924915	40714879876	21614153083	6637513733	899222457								185403189015
9	1622632573	22032243199	131454037507	456089759102	1022580411854	1551962730317	1617784266946	1145410405600	527760931864	143036657697	17329515172							6637063591831
10	28091567595	419653154944	2782597410493	10863485753033	27841986315406	49308782855256	61693515010126	54611698087145	33558068661996	13645687313358	3307212498292	362164300173						258422942927817
11	526858348381	8587492161897	62645744477746	271795854408935	783795925170116	1586707768268050	2316529992976200	2461189891849402	1890805579669007	1025333378011084	372824583893334	81669516021196	8155216185781					10870567801441129
12	10641342970443	187927249132390	1495770670493436	7139575215585099	22879941085172170	52115552503858559	86969530737322804	107817355760982726	99412232251276801	67396400086828242	32680142269964474	10739895313686540	2145236888760679	196789115887252				491186990491921615
13	230283190977853	4380165950951891	37772925307660661	196712024634442576	693524382475706783	1755449097957580310	3296081003435761971	4670563948671567660	5024880312853334077	4088488025851231573	2478444705210924165	1086159599493251282	325476732278228727	59722204732589323	5064722539020379			23722950134583229231
14	5315654681981355	108340770975518284	1006264904664679559	5677692183665625449	21839336348059856677	60814574419404275042	126889155154394204285	202301323200032525184	248769862378586254419	236174300193205117316	171823191497392436143	94136462891117205339	37616191762604272297	10354618219398579618	1756451730488802961	138457553073641465		1219411538861744975393
15	130370767029135901	2834537863008533821	28210179361807213900	171431759805611819425	714435736536801175651	2170515486977152997395	4982139877981253232246	8826562692850942253499	12212021398100837763594	13251318421598345116885	11248640156889753029556	7396097207486890305337	3694664719728606893257	1356184209842959753871	345110271699363791142	54414454073297521927	4006059432756066914	66458717540996416604321

Technical details

Node id: n-3068f5c4614a7a59 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-e6f2903ba541c565

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:          13,           35,            23;
   2:          75,          304,           401,            173;
   3:         541,         2929,          5806,           5019,           1602;
   4:        4683,        31726,         83944,         108971,          69644,          17575;
   5:       47293,       384747,       1273441,        2205804,        2116241,        1068920,         222497;
   6:      545835,      5183284,      20597667,       44622201,       57112019,       43295414,       18034183,        3188806;
   7:     7087261,     76944533,     356868968,      928109741,     1485477001,     1502110525,      938928838,      332185673,       50988405;
   8:   102247563,   1249200538,    6623511626,    20103093515,    38623441709,    48835924915,    40714879876,    21614153083,     6637513733,      899222457;
   9:  1622632573,  22032243199,  131454037507,   456089759102,  1022580411854,  1551962730317,  1617784266946,  1145410405600,   527760931864,   143036657697,   17329515172;
  10: 28091567595, 419653154944, 2782597410493, 10863485753033, 27841986315406, 49308782855256, 61693515010126, 54611698087145, 33558068661996, 13645687313358, 3307212498292, 362164300173;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = -(4*n-3-(2*n^2-17*n+7)*x-(6*n^2-24*n+4)*x^2-(6*n^2-13*n)*x^3-(2*n^2-2*n)*x^4)*P(n-1,x) + (4-(4*n-23)*x-(16*n-49)*x^2-(24*n-49)*x^3-(16*n-23)*x^4-(4*n-4)*x^5)*d/dx P(n-1,x) + (2*x+10*x^2+20*x^3+20*x^4+10*x^5+2*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 13 + 35*x + 23*x^2.

%t Mathematica

P[1, x_] := 13 + 35*x + 23*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 3 - (2*n^2 - 17*n + 7)*x - (6*n^2 - 24*n + 4)*x^2 - (6*n^2 - 13*n)*x^3 - (2*n^2 - 2*n)*x^4)*P[n - 1, x] + (4 - (4*n - 23)*x - (16*n - 49)*x^2 - (24*n - 49)*x^3 - (16*n - 23)*x^4 - (4*n - 4)*x^5)*D[P[n - 1, x], x] + (2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 13 + 35*x + 23*x^2; for(n = 2, N, my(i = n); v[i] = (-(4*n - 3 - (2*n^2 - 17*n + 7)*x - (6*n^2 - 24*n + 4)*x^2 - (6*n^2 - 13*n)*x^3 - (2*n^2 - 2*n)*x^4)*v[i - 1] + (4 - (4*n - 23)*x - (16*n - 49)*x^2 - (24*n - 49)*x^3 - (16*n - 23)*x^4 - (4*n - 4)*x^5)*deriv(v[i - 1], x) + (2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6)*deriv(deriv(v[i - 1], x), 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 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.?
  - 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.