A257622 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	8	4							12
2	88	88	16						192
3	1232	1848	744	64					3888
4	20944	41888	26664	5720	256				95472
5	418880	1047200	917840	329560	42088	1024			2756592
6	9634240	28902720	32358480	16545760	3758568	302808	4096		91506672
7	250490240	876715840	1196321280	799013600	267714832	40916568	2152424	16384	3433341168

All 15 rows

Technical details

Node id: n-4c6e90a40d8a7363 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-31113a40d89e3fd5

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:            8,             4;
  2:           88,            88,            16;
  3:         1232,          1848,           744,            64;
  4:        20944,         41888,         26664,          5720,           256;
  5:       418880,       1047200,        917840,        329560,         42088,         1024;
  6:      9634240,      28902720,      32358480,      16545760,       3758568,       302808,        4096;
  7:    250490240,     876715840,    1196321280,     799013600,     267714832,     40916568,     2152424,      16384;
  8:   7264216960,   29056867840,   46803137920,   38710376320,   17335369744,   4053124768,   432842344,   15198040,     65536;
  9: 232454942720, 1046047242240, 1943303281920, 1920007689600, 1082204544960, 346269036960, 58750414800, 4495601880, 106910568, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (3*n+5+(3*n+1)*x)*P(n-1,x) - (3*x+3*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[(3*n + 5 + (3*n + 1)*x)*P[n - 1, x] - (3*x + 3*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] = (3*n + 5 + (3*n + 1)*x)*v[i - 1] - (3*x + 3*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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.?
  - 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257622 → 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