A257615 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 3\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	10	5														15
2	120	120	25													265
3	1680	2520	1090	125												5415
4	26880	53760	35760	8880	625											125905
5	483840	1209600	1092000	428400	68410	3125										3285375
6	9676800	29030400	33196800	18009600	4676520	510120	15625									95115865
7	212889600	745113600	1032998400	719712000	263576880	48210120	3727090	78125								3026305815
8	5109350400	20437401600	33326899200	28449792000	13573486080	3574287360	478616160	26870880	390625							104977094305
9	132843110400	597793996800	1121502412800	1135553126400	671069629440	233643916800	46017787200	4629996000	192002410	1953125						3943247931375
10	3719607091200	18598035456000	39469731840000	46290714624000	32725795737600	14271634944000	3773655513600	571015603200	43973992920	1363548120	9765625					159465538116265
11	111588212736000	613735170048000	1453967843328000	1939841519616000	1599491084083200	841771226419200	282000323404800	58193771328000	6896807536080	412128513720	9642493090	48828125				6907907778334215
12	3570822807552000	21424936845312000	56054478864384000	83877139906560000	79124457929932800	48909297578803200	19950117317222400	5277492734976000	866867947841280	81634682088960	3824866540560	67985732880	244140625			319141139711086705
13	121407975456768000	789151840468992000	2260107660754944000	3749921888993280000	3986410124029132800	2835436758419865600	1368179234976153600	444701849092300800	94766014085783040	12575438235360000	951529634546400	35239627659600	478341536410	1220703125		15663646033337025375
14	4370687116443648000	30594809815105536000	95208848517758976000	173520563510181888000	205367530697647718400	165408788571586560000	92380473566244864000	35778846144577536000	9455251716284774400	1647848099523225600	178705171212422400	10960180767244800	322896747373320	3360597786120	6103515625	813923639370783068665

Technical details

Node id: n-98b23464d77c0584 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-77f8f710632b75b3

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:           10,            5;
  2:          120,          120,            25;
  3:         1680,         2520,          1090,           125;
  4:        26880,        53760,         35760,          8880,          625;
  5:       483840,      1209600,       1092000,        428400,        68410,         3125;
  6:      9676800,     29030400,      33196800,      18009600,      4676520,       510120,       15625;
  7:    212889600,    745113600,    1032998400,     719712000,    263576880,     48210120,     3727090,      78125;
  8:   5109350400,  20437401600,   33326899200,   28449792000,  13573486080,   3574287360,   478616160,   26870880,    390625;
  9: 132843110400, 597793996800, 1121502412800, 1135553126400, 671069629440, 233643916800, 46017787200, 4629996000, 192002410, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+8+(2*n+3)*x)*P(n-1,x) - (2*x+2*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[(2*n + 8 + (2*n + 3)*x)*P[n - 1, x] - (2*x + 2*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] = (2*n + 8 + (2*n + 3)*x)*v[i - 1] - (2*x + 2*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) = 2*x + 5.?
  - 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) = 2*x + 5.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257615 → 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