A257606 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	8	4														12
2	72	72	16													160
3	720	1080	488	64												2352
4	7920	15840	10872	2952	256											37840
5	95040	237600	219600	91800	16808	1024										661872
6	1235520	3706560	4316400	2455200	702072	92232	4096									12512080
7	17297280	60540480	85155840	61538400	24207120	5042520	493928	16384								254291952
8	259459200	1037836800	1712430720	1504863360	758585520	219875040	34700472	2600712	65536							5530417360
9	4151347200	18681062400	35390234880	36687530880	22646796480	8487309600	1886130000	231609240	13527848	262144						128175810672
10	70572902400	352864512000	755026272000	902918016000	661283663040	305668883520	88645906800	15519002400	1511406072	69736392	1048576					3154081349200
11	1270312243200	6986717337600	16667659008000	22604085504000	19190069458560	10586532116160	3814770833280	879876280800	123747077520	9696063960	357070568	4194304				82133827187952
12	24135932620800	144815595724800	381164245862400	578344935168000	559100067926400	359302213670400	155457341994240	44891452869120	8400228099120	963190182240	61390018872	1818907272	16777216			2256638429820880
13	482718652416000	3137671240704000	9033190361395200	15168163339929600	16464085244006400	12098734020172800	6128727138293760	2138162307240960	504825813011520	77796916797600	7356231464400	384710278680	9228754088	67108864		65241825271573872
14	10137091700736000	70959641905152000	221833356717772800	408524795539660800	492243735907123200	407612852494848000	237115715316480000	97350270822604800	27944358641559360	5476995318674880	703293880489200	55340723037600	2391320458872	46680641352	268435456	1979959887237674320

Technical details

Node id: n-8e2b4e2d6609e6fb · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-79e4efb037c60552

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:         72,          72,          16;
  3:        720,        1080,         488,          64;
  4:       7920,       15840,       10872,        2952,         256;
  5:      95040,      237600,      219600,       91800,       16808,       1024;
  6:    1235520,     3706560,     4316400,     2455200,      702072,      92232,       4096;
  7:   17297280,    60540480,    85155840,    61538400,    24207120,    5042520,     493928,     16384;
  8:  259459200,  1037836800,  1712430720,  1504863360,   758585520,  219875040,   34700472,   2600712,    65536;
  9: 4151347200, 18681062400, 35390234880, 36687530880, 22646796480, 8487309600, 1886130000, 231609240, 13527848, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (n+7+(n+3)*x)*P(n-1,x) - (x+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[(n + 7 + (n + 3)*x)*P[n - 1, x] - (x + 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] = (n + 7 + (n + 3)*x)*v[i - 1] - (x + 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) = 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) = x + 4.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257606 → 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