A142461 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(6\,x + 18\,x^{2} + 18\,x^{3} + 6\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	16	16													33
3	1	114	336	224												675
4	1	800	5280	8960	4480											19521
5	1	5602	75040	244160	291200	116480										732483
6	1	39216	1020336	5689600	12163200	11182080	3727360									33821793
7	1	274514	13578096	122387104	417872000	662887680	495738880	141639680								1854377955
8	1	1921600	178711360	2515448320	12894542080	31414190080	39554744320	24928583680	6232145920							117720287361
9	1	13451202	2338620480	50295477120	372672518400	1309097986560	2468779622400	2575009382400	1402232832000	311607296000						8492047186563
10	1	94158416	30509675856	988354752000	10322722502400	50271523061760	133235981598720	204539029094400	182009821593600	87250042880000	17450008576000					686098087893153
11	1	659108914	397379053456	19205875749984	277835157600000	1826807999976960	6538420057128960	13858145551810560	17918198538240000	13899243438080000	5950452924416000	1081900531712000				61370608112876835
12	1	4613762400	5171200566240	370474945998080	7329996454999680	63854762096885760	300379398113034240	844360220898754560	1487750132653424640	1660368316006400000	1141335260921856000	441415416938496000	73569236156416000			6020738391000593601
13	1	32296336802	67262517460320	7111420781890880	190659410294953600	2170077532833450240	13157390117282611200	47721906626941747200	110051606219563008000	165707764013023232000	162602076612591616000	100337619112034304000	35386802591236096000	5444123475574784000		642777204426971490243
14	1	226074357616	874671097678576	136058670100371200	4908713673011657600	72229548185505751040	556265915390127022080	2552022809415234355200	7492292596344024268800	14616507331694428160000	19198361458097389568000	16803949230486978560000	9406194688778567680000	3048709146321879040000	435529878045982720000	74187108250000401190113

Technical details

Node id: n-552c7386b5928d1b · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-3980a5ccbbfbe4b6

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,        2;
  2: 1,       16,         16;
  3: 1,      114,        336,         224;
  4: 1,      800,       5280,        8960,         4480;
  5: 1,     5602,      75040,      244160,       291200,        116480;
  6: 1,    39216,    1020336,     5689600,     12163200,      11182080,       3727360;
  7: 1,   274514,   13578096,   122387104,    417872000,     662887680,     495738880,     141639680;
  8: 1,  1921600,  178711360,  2515448320,  12894542080,   31414190080,   39554744320,   24928583680,    6232145920;
  9: 1, 13451202, 2338620480, 50295477120, 372672518400, 1309097986560, 2468779622400, 2575009382400, 1402232832000, 311607296000;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (6*x+18*x^2+18*x^3+6*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[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*D[P[n - 1, x], 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 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*deriv(v[i - 1], x))/(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: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142461 → x ↦ x+1

References

[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