A094262 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 9\,x + 33\,x^{2} + 63\,x^{3} + 66\,x^{4} + 36\,x^{5} + 8\,x^{6} - 2\,n x - 14\,n x^{2} - 38\,n x^{3} - 50\,n x^{4} - 32\,n x^{5} - 8\,n x^{6}\right)\,P_{n-1}(x) + \left(x + 8\,x^{2} + 26\,x^{3} + 44\,x^{4} + 41\,x^{5} + 20\,x^{6} + 4\,x^{7}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	4	4																											9
2	1	10	36	56	32																									135
3	1	20	146	528	1024	1024	416																							3159
4	1	38	478	3016	11000	24352	32416	23936	7552																					102789
5	1	72	1444	14140	81376	298576	721984	1150528	1166080	682752	176128																			4293081
6	1	138	4224	60280	502668	2714832	10005408	25773120	46552320	57933056	47408640	23000064	5018624																	218973375
7	1	268	12214	244552	2810956	20946448	107987904	399359232	1078851456	2138152448	3080602112	3144452096	2157759488	893702144	168968192															13193849511
8	1	526	35250	966360	14823900	146354520	1003132640	4990836320	18507620160	51888892160	110486117888	177828123648	213045027840	184310446080	108896133120	39348527104	6563282944													917030320461
9	1	1040	102008	3767524	75436020	959955360	8459217320	54344440480	262880626240	977013635200	2822041111168	6362832824320	11173028056064	15134028214272	15514954168320	11646491336704	6041406586880	1936045801472	288909131776											72223474412169
10	1	2066	296596	14599592	375572488	6040048712	66832282040	538999833760	3288056861120	15546856917120	57887197027968	171340303385088	404817512992768	762787026507776	1139245217941504	1331853637181440	1193117302636544	790959941976064	365675024580608	105276955820032	14212910809088									6356620207272375
11	1	4116	866986	56478224	1845310072	36962048528	504668450960	5006172225776	37599801771680	219859329368640	1020133344215168	3804234182569728	11492386584683008	28231584702292992	56393341905995776	91232776290193408	118517149842202624	121856461129695232	96929088456851456	57533210611744768	23989733004541952	6269532953640960	772776684683264							618305418529835319
12	1	8214	2547366	218664104	8997814896	222004858992	3691165027648	44346206757312	402821660248848	2853317329480192	16105302310339968	73551176354084352	274644471851929088	844172958596739072	2143306944580835328	4497114430413414400	7777736319943802880	11021272596006174720	12666272433301258240	11621682963659489280	8312457695016910848	4466480737641562112	1696044411718729728	405818359060365312	46017323176296448					65865982531217339349
13	1	16408	7518796	848492684	43701398456	1316935826672	26389942810000	379417226475712	4111304413099408	34727379174524992	234196710104279488	1282926252713681152	5780206638975139328	21608098659636134912	67411304022561118208	176066318969924669440	385285339825142628352	705234012193081507840	1075144282996219248640	1355121161590347857920	1396238367713309360128	1156569631249806393344	751473348157023715328	368836850612622589952	128573138922816667648	28368666057433415680	2978458881388183552			7626245555305356754041
14	1	32794	22277352	3302235544	211934323828	7749004914720	185630650728080	3163595481621248	40451264026518480	402471717791307936	3198050864077416384	20684242380854142336	110433856660275000064	491777529036723122176	1840221400984049147904	5815544519457665122304	15566680200403086565376	35326345715649299300352	67898555625519608856576	110189245321495095214080	150162774530957602848768	170412807915952943398912	159110075939614297686016	120118753152388439212032	71496737414981992054784	32296474812877220347904	10405077567260785115136	2129955053550732050432	208198894960190160896	953603988025167150722415

Technical details

Node id: n-56cb29b4e81fea25 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-3611becd7eab4c6b

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,    4,      4;
  2: 1,   10,     36,      56,       32;
  3: 1,   20,    146,     528,     1024,      1024,        416;
  4: 1,   38,    478,    3016,    11000,     24352,      32416,       23936,         7552;
  5: 1,   72,   1444,   14140,    81376,    298576,     721984,     1150528,      1166080,       682752,        176128;
  6: 1,  138,   4224,   60280,   502668,   2714832,   10005408,    25773120,     46552320,     57933056,      47408640,      23000064,        5018624;
  7: 1,  268,  12214,  244552,  2810956,  20946448,  107987904,   399359232,   1078851456,   2138152448,    3080602112,    3144452096,     2157759488,      893702144,      168968192;
  8: 1,  526,  35250,  966360, 14823900, 146354520, 1003132640,  4990836320,  18507620160,  51888892160,  110486117888,  177828123648,   213045027840,   184310446080,   108896133120,    39348527104,    6563282944;
  9: 1, 1040, 102008, 3767524, 75436020, 959955360, 8459217320, 54344440480, 262880626240, 977013635200, 2822041111168, 6362832824320, 11173028056064, 15134028214272, 15514954168320, 11646491336704, 6041406586880, 1936045801472, 288909131776;

%F Formula

The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(2*n-9)*x-(14*n-33)*x^2-(38*n-63)*x^3-(50*n-66)*x^4-(32*n-36)*x^5-(8*n-8)*x^6)*P(n-1,x) + (x+8*x^2+26*x^3+44*x^4+41*x^5+20*x^6+4*x^7)*d/dx 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[((1 - (2*n - 9)*x - (14*n - 33)*x^2 - (38*n - 63)*x^3 - (50*n - 66)*x^4 - (32*n - 36)*x^5 - (8*n - 8)*x^6)*P[n - 1, x] + (x + 8*x^2 + 26*x^3 + 44*x^4 + 41*x^5 + 20*x^6 + 4*x^7)*D[P[n - 1, x], x])/(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] = ((1 - (2*n - 9)*x - (14*n - 33)*x^2 - (38*n - 63)*x^3 - (50*n - 66)*x^4 - (32*n - 36)*x^5 - (8*n - 8)*x^6)*v[i - 1] + (x + 8*x^2 + 26*x^3 + 44*x^4 + 41*x^5 + 20*x^6 + 4*x^7)*deriv(v[i - 1], 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) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - 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.