A257610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	28	28	4													60
3	280	420	156	8												864
4	3640	7280	4452	812	16											16200
5	58240	145600	124600	41300	4124	32										373896
6	1106560	3319680	3658200	1783600	359268	20748	64									10248120
7	24344320	85205120	114965760	74401600	23212280	3019380	103996	128								325252584
8	608608000	2434432000	3888451840	3144843520	1343382040	285528880	24883012	520492	256							11730650040
9	17041024000	76684608000	141537876480	137521063680	74956452480	22519224000	3389647800	202707540	2603484	512						473855207976
10	528271744000	2641358720000	5532246720000	6280834560000	4174593870720	1634567276160	359334557400	39313201200	1639884708	13019468	1024					21192173554680
11	17961239296000	98786816128000	231502311040000	300859278720000	236300260035840	114548655661440	33536996613120	5541755419200	448844060280	13210213940	65101436	2048				1039499432291304
12	664565853952000	3987395123712000	10337749758336000	15137626824320000	13729847747616000	7954976405376000	2928176048878080	658797029130240	83419548652440	5069386802480	106137421572	325515372	4096			55487730189716280
13	26582634158080000	172787122027520000	491042321113344000	800074423820672000	823426914838528000	555850451572416000	247384330398965760	71080664531036160	12534262273653120	1233775709566400	56824629043000	851377980180	1627593244	8192		3202054578078406056
14	1143053268797440000	8001372881582080000	24734476518239488000	44389011648869888000	51128615922255872000	39443193174344960000	20636720000856960000	7253614218859008000	1659764273820132480	232822870005165440	18011580111488600	633584699274800	6822416994148	8137998988	16384	198641296772996766840

Technical details

Node id: n-30bdb3d8de09da44 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-dbcd952a44050fb4

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:           4,           2;
  2:          28,          28,            4;
  3:         280,         420,          156,            8;
  4:        3640,        7280,         4452,          812,          16;
  5:       58240,      145600,       124600,        41300,        4124,          32;
  6:     1106560,     3319680,      3658200,      1783600,      359268,       20748,         64;
  7:    24344320,    85205120,    114965760,     74401600,    23212280,     3019380,     103996,       128;
  8:   608608000,  2434432000,   3888451840,   3144843520,  1343382040,   285528880,   24883012,    520492,     256;
  9: 17041024000, 76684608000, 141537876480, 137521063680, 74956452480, 22519224000, 3389647800, 202707540, 2603484, 512;

%F Formula

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

Related ?

Same fingerprint ?

A257610 → 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