A049374 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 16\,x + 16\,x^{2} + 5\,x^{3} + n - 3\,n x - 9\,n x^{2} - 5\,n x^{3}\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	7														13
2	42	102	61													205
3	336	1284	1596	649												3865
4	3024	16296	31704	26676	8245											85945
5	30240	217224	589896	771204	490506	122215										2221285
6	332640	3082464	10980144	19836096	19467714	9931362	2069425									65699845
7	3991680	46747008	210006720	491453424	661233216	517676376	219973656	39328465								2190410545
8	51891840	757620864	4176228672	12148278912	20993608944	22406064912	14552983872	5289975144	827226505							81203879665
9	726485760	13097500416	86789501568	304902214848	649237238496	885033692928	780610734288	432472243896	137189274030	19047582055						3309106468285
10	10897286400	240881921280	1888193172096	7842060914688	19990573007328	33395890895136	37482259632048	28121871099168	13567611057066	3814785459870	475956135205					146830980580285
11	174356582400	4698909895680	43007873965056	207717690660480	620606718635904	1234693405737792	1692196898071680	1611339163744464	1050650257223952	448551566246652	113179765565460	12815133759385				7039631740088905
12	2964061900800	96924824156160	1024752417841152	5680886748945408	19573070426477568	45410235570475776	73863033331703616	85669362998227008	70756334613540576	40761374893033080	15596523445599096	3567499198833180	369605936607805			362372568467341225
13	53353114214400	2107744417658880	25510554436663296	160618596611945472	630049301075690496	1677427155401437440	3171194902996952832	4354515627241747968	4372387934582828160	3185729593233491304	1642744271788547928	569201701409672604	119021473318907250	11361372997850695		19921923582627608725
14	1013709170073600	48184975884165120	662567071621238784	4696803758830030848	20759563298756265984	62625618501687521280	135423467475451803648	215614681199951185152	255841957216537211520	226284802860089990016	147397488084346211712	68757268988016082656	21759642906994173090	4188948816640776090	370609338222772825	1164432618202199502325

Technical details

Node id: n-b078e2dab7ab1a26 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-c2d44d94208eac35

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:         6,           7;
  2:        42,         102,          61;
  3:       336,        1284,        1596,          649;
  4:      3024,       16296,       31704,        26676,         8245;
  5:     30240,      217224,      589896,       771204,       490506,       122215;
  6:    332640,     3082464,    10980144,     19836096,     19467714,      9931362,      2069425;
  7:   3991680,    46747008,   210006720,    491453424,    661233216,    517676376,    219973656,     39328465;
  8:  51891840,   757620864,  4176228672,  12148278912,  20993608944,  22406064912,  14552983872,   5289975144,    827226505;
  9: 726485760, 13097500416, 86789501568, 304902214848, 649237238496, 885033692928, 780610734288, 432472243896, 137189274030, 19047582055;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (n+5-(3*n-16)*x-(9*n-16)*x^2-(5*n-5)*x^3)*P(n-1,x) + (5*x+15*x^2+15*x^3+5*x^4)*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[((n + 5 - (3*n - 16)*x - (9*n - 16)*x^2 - (5*n - 5)*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*D[P[n - 1, x], x])/(1 + x)]];
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] = ((n + 5 - (3*n - 16)*x - (9*n - 16)*x^2 - (5*n - 5)*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*deriv(v[i - 1], x))/(1 + x)); 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 A triangle of numbers related to triangle A030527.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.