A200536 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 17\,x^{2} + 17\,x^{3} + 6\,x^{4}\right)\,P_{n-1}(x)\)

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

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	5	6																											12
2	1	10	37	60	36																									144
3	1	15	93	305	558	540	216																							1728
4	1	20	174	860	2641	5160	6264	4320	1296																					20736
5	1	25	280	1850	7985	23525	47910	66600	60480	32400	7776																			248832
6	1	30	411	3400	18915	74550	213445	447300	680940	734400	532656	233280	46656																	2985984
7	1	35	567	5635	38381	189525	699685	1961825	4198110	6822900	8290296	7302960	4408992	1632960	279936															35831808
8	1	40	748	8680	69958	415240	1877596	6597400	18205345	39584400	67593456	89691840	90665568	67495680	34898688	11197440	1679616													429981696
9	1	45	954	12660	117846	817110	4373544	18476820	62457921	170195525	374747526	665165520	944685504	1058974560	916370496	590664960	267058944	75582720	10077696											5159780352
10	1	50	1185	17700	186870	1482300	9166170	45247200	181083285	593346050	1600472677	3560076300	6518998260	9773395200	11879356320	11526364800	8718606720	4954867200	1990344960	503884800	60466176									61917364224
11	1	55	1441	23925	282480	2522850	17698890	99971850	462316305	1770245675	5653702637	15122515985	33922215822	63728844300	99860321880	129563517600	137626568640	117706089600	79076321280	40184812800	14521959936	3325639680	362797056							743008370688
12	1	60	1722	31460	410751	4078800	32008020	203603400	1068368895	4681658300	17278828842	54012503220	143457011569	324075019320	622037838312	1011238192800	1384606087920	1583220038400	1493366181120	1141802956800	689903951616	317044316160	104122755072	21767823360	2176782336					8916100448256
13	1	65	2028	40430	578383	6321315	54866526	388116300	2278434015	11245123175	47097333712	168496597230	517192500721	1365435096485	3103155004326	6065877500280	10173024081792	14573679634800	17717102900640	18107954092800	15359115822336	10617381815040	5828768045568	2444647495680	735752429568	141490851840	13060694016			106993205379072
14	1	70	2359	50960	792701	9455810	89943399	700376820	4548214671	24965991050	116993553677	471454004840	1642259489143	4962377183470	13033485491077	29774263100820	59121341609148	101834065045440	151623645565392	194135546404800	212201503690176	196060685483520	151070372054784	95292778613760	47931598181376	18488137973760	5135029530624	914248581120	78364164096	1283918464548864

Technical details

Node id: n-597121562fed1ded · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-9f63f55d8b2eb3fe

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  5,   6;
  2: 1, 10,  37,    60,     36;
  3: 1, 15,  93,   305,    558,    540,     216;
  4: 1, 20, 174,   860,   2641,   5160,    6264,     4320,     1296;
  5: 1, 25, 280,  1850,   7985,  23525,   47910,    66600,    60480,     32400,      7776;
  6: 1, 30, 411,  3400,  18915,  74550,  213445,   447300,   680940,    734400,    532656,    233280,     46656;
  7: 1, 35, 567,  5635,  38381, 189525,  699685,  1961825,  4198110,   6822900,   8290296,   7302960,   4408992,    1632960,    279936;
  8: 1, 40, 748,  8680,  69958, 415240, 1877596,  6597400, 18205345,  39584400,  67593456,  89691840,  90665568,   67495680,  34898688,  11197440,   1679616;
  9: 1, 45, 954, 12660, 117846, 817110, 4373544, 18476820, 62457921, 170195525, 374747526, 665165520, 944685504, 1058974560, 916370496, 590664960, 267058944, 75582720, 10077696;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+7*x+17*x^2+17*x^3+6*x^4)*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 + 7*x + 17*x^2 + 17*x^3 + 6*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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 + 7*x + 17*x^2 + 17*x^3 + 6*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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 of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Parent-row provers of polytool polytool ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.