A142461 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

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

Technical details

Node id: n-b97a03f73769dd58 · revision 0 · JSON
Definition source: derived:shift_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:            2,             1;
  2:           16,            16,             1;
  3:          224,           336,           114,             1;
  4:         4480,          8960,          5280,           800,             1;
  5:       116480,        291200,        244160,         75040,          5602,            1;
  6:      3727360,      11182080,      12163200,       5689600,       1020336,        39216,           1;
  7:    141639680,     495738880,     662887680,     417872000,     122387104,     13578096,      274514,          1;
  8:   6232145920,   24928583680,   39554744320,   31414190080,   12894542080,   2515448320,   178711360,    1921600,        1;
  9: 311607296000, 1402232832000, 2575009382400, 2468779622400, 1309097986560, 372672518400, 50295477120, 2338620480, 13451202, 1;

%F Formula

The row polynomials satisfy P(n,x) = (6*n-4+(6*n-5)*x)*P(n-1,x) - (6*x+6*x^2)*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[(6*n - 4 + (6*n - 5)*x)*P[n - 1, x] - (6*x + 6*x^2)*D[P[n - 1, x], 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] = (6*n - 4 + (6*n - 5)*x)*v[i - 1] - (6*x + 6*x^2)*deriv(v[i - 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]
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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/(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