A151775 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(3 + 36\,x + 146\,x^{2} + 284\,x^{3} + 291\,x^{4} + 152\,x^{5} + 32\,x^{6} - 2\,n - 34\,n x - 162\,n x^{2} - 350\,n x^{3} - 388\,n x^{4} - 216\,n x^{5} - 48\,n x^{6} + 4\,n^{2} x + 28\,n^{2} x^{2} + 76\,n^{2} x^{3} + 100\,n^{2} x^{4} + 64\,n^{2} x^{5} + 16\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(2 + 40\,x + 228\,x^{2} + 616\,x^{3} + 914\,x^{4} + 768\,x^{5} + 344\,x^{6} + 64\,x^{7} - 8\,n x - 64\,n x^{2} - 208\,n x^{3} - 352\,n x^{4} - 328\,n x^{5} - 160\,n x^{6} - 32\,n x^{7}\right)\,P_{n-1}'(x) + \left(4\,x + 36\,x^{2} + 136\,x^{3} + 280\,x^{4} + 340\,x^{5} + 244\,x^{6} + 96\,x^{7} + 16\,x^{8}\right)\,P_{n-1}''(x)\)

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

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	3														4
2	5	38	57													100
3	61	845	2827	2763												6496
4	1385	30108	165678	347372	250737											795280
5	50521	1578727	12558738	42044902	63967093	36581523										156781504
6	2702765	114545698	1224886319	5787668396	13858225819	16532652626	7828053417									45348735040
7	199360981	10991602449	150729398853	938210069225	3131754984615	5828199544491	5717183815999	2309644635483								18086913412096
8	19391512145	1347876178808	22928729094540	179838785780392	782298317080150	2013821093811720	3063007488423788	2550944489331800	898621108880097							9512827280093440
9	2404879675441	205627071360011	4232829305684692	40549809708954860	220137003799232878	733364018069036906	1534595693858402596	1972903207941495932	1427429993305584745	445777636063460643						6379198224002888704
10	370371188237525	38192963484825438	933194270507210901	10668571399857360840	70126483754691692874	289725140436733413972	780737836262821173618	1375383343626576431496	1530969478821186570945	979132856617134896894	274613643571568682777					5312329112095750497280
11	69348874393137901	8485430262906491413	242310663323811617775	3246874631238340139655	25267623599240044803570	125671421140431564460482	418100696673883999361166	945041685519872016609150	1437007969196408485533105	1410173986838150762956345	808076117539129224076531	205676334188681975553003				5378513574769497524740096
12	15514534163557086905	2221953397527468445588	73216705496382307863962	1133456630939706460747380	10263279419149150772777775	60113750262643983637864680	239764795173431468737080396	666823308974283903810195144	1297276706872288093281033255	1734783092851375735468573700	1522048307327010116269406650	790003354483354014418275908	184053312545818735778213457			6506338818713722981967564800
13	4087072509293123892361	677218227067978981388655	25476537149172380969399622	450302324839434940830900178	4678932635022444855733896555	31730456184340300468784606997	148467544207973087090004913380	493467613407499454107701542604	1178109511071432251200013266191	2011040951076885366056843008105	2399575876141725070626165508086	1903560179146593811375011283650	902933937813209930220954062557	193944394596325636374396208563		9267985856448295536969513877504
14	1252259641403629865468285	237677451783460555460663258	10115394887934990179968610283	202125732022428439089904017812	2382852990461812161446710833677	18458312051467481841096003568470	99638756041382566995621376906851	387367970628116634570361025460696	1102498341235266613052456425290879	2304582962762315547567960383722214	3503617610484824447538931020111785	3773614105062190406263529186678868	2731680572490898664793965751078263	1192992219946978399215416496012842	237694572315172714282278845824137	15354740756065697076576518424248320

Technical details

Node id: n-51aaefce50b145f1 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-50514a51a60708e3

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:             1;
  1:             1,               3;
  2:             5,              38,               57;
  3:            61,             845,             2827,              2763;
  4:          1385,           30108,           165678,            347372,             250737;
  5:         50521,         1578727,         12558738,          42044902,           63967093,           36581523;
  6:       2702765,       114545698,       1224886319,        5787668396,        13858225819,        16532652626,          7828053417;
  7:     199360981,     10991602449,     150729398853,      938210069225,      3131754984615,      5828199544491,       5717183815999,       2309644635483;
  8:   19391512145,   1347876178808,   22928729094540,   179838785780392,    782298317080150,   2013821093811720,    3063007488423788,    2550944489331800,     898621108880097;
  9: 2404879675441, 205627071360011, 4232829305684692, 40549809708954860, 220137003799232878, 733364018069036906, 1534595693858402596, 1972903207941495932, 1427429993305584745, 445777636063460643;

%F Formula

The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -(2*n-3-(4*n^2-34*n+36)*x-(28*n^2-162*n+146)*x^2-(76*n^2-350*n+284)*x^3-(100*n^2-388*n+291)*x^4-(64*n^2-216*n+152)*x^5-(16*n^2-48*n+32)*x^6)*P(n-1,x) + (2-(8*n-40)*x-(64*n-228)*x^2-(208*n-616)*x^3-(352*n-914)*x^4-(328*n-768)*x^5-(160*n-344)*x^6-(32*n-64)*x^7)*d/dx P(n-1,x) + (4*x+36*x^2+136*x^3+280*x^4+340*x^5+244*x^6+96*x^7+16*x^8)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 3 - (4*n^2 - 34*n + 36)*x - (28*n^2 - 162*n + 146)*x^2 - (76*n^2 - 350*n + 284)*x^3 - (100*n^2 - 388*n + 291)*x^4 - (64*n^2 - 216*n + 152)*x^5 - (16*n^2 - 48*n + 32)*x^6)*P[n - 1, x] + (2 - (8*n - 40)*x - (64*n - 228)*x^2 - (208*n - 616)*x^3 - (352*n - 914)*x^4 - (328*n - 768)*x^5 - (160*n - 344)*x^6 - (32*n - 64)*x^7)*D[P[n - 1, x], x] + (4*x + 36*x^2 + 136*x^3 + 280*x^4 + 340*x^5 + 244*x^6 + 96*x^7 + 16*x^8)*D[P[n - 1, x], {x, 2}])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 3 - (4*n^2 - 34*n + 36)*x - (28*n^2 - 162*n + 146)*x^2 - (76*n^2 - 350*n + 284)*x^3 - (100*n^2 - 388*n + 291)*x^4 - (64*n^2 - 216*n + 152)*x^5 - (16*n^2 - 48*n + 32)*x^6)*v[i - 1] + (2 - (8*n - 40)*x - (64*n - 228)*x^2 - (208*n - 616)*x^3 - (352*n - 914)*x^4 - (328*n - 768)*x^5 - (160*n - 344)*x^6 - (32*n - 64)*x^7)*deriv(v[i - 1], x) + (4*x + 36*x^2 + 136*x^3 + 280*x^4 + 340*x^5 + 244*x^6 + 96*x^7 + 16*x^8)*deriv(deriv(v[i - 1], x), x))/(1 + 3*x + 3*x^2 + x^3)); 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.?
  - 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.