A257617 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 5\,x + 7\,n + 7\,n x\right)\,P_{n-1}(x) + \left(-7\,x - 7\,x^{2}\right)\,P_{n-1}'(x)\)

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

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	44	44	4													92
3	792	1188	412	8												2400
4	19800	39600	23540	3740	16											86696
5	633600	1584000	1334520	417780	33724	32										4003656
6	24710400	74131200	80883000	38214000	7055444	303644	64									225297752
7	1136678400	3978374400	5331110400	3381840000	1005919992	116227188	2733052	128								14952883560
8	60243955200	240975820800	382961779200	305469964800	126603199800	25228249200	1889698580	24597980	256							1143397265816
9	3614637312000	16265867904000	29906008704000	28763646912000	15353691091200	4428802224000	614264306040	30505755060	221382844	512						98977645591656
10	242180699904000	1210903499520000	2528438799744000	2848334201856000	1864394416684800	709808241542400	148220674371000	14677182630000	490527292244	1992447644	1024					9567450235993112
11	17921371792896000	98567544860928000	230415055453440000	297611163083520000	231027187462156800	109715875754284800	31005966516940800	4813549796880000	346404691750392	7870353599988	17932032892	2048				1021432007698431720
12	1451631115224576000	8709786691347456000	22534671945367296000	32833648389484800000	29505888394338470400	16829316845267097600	6036731227349222400	1301254354626969600	153066611200159800	8108974275058800	126122909961620	161388304220	4096			119364230832779376536
13	127743538139762688000	830332997908457472000	2355638872573628928000	3822350822161926912000	3904540448976820224000	2603295319670787456000	1135984795566686208000	316365093833817600000	53044542679602988800	4794722692881264000	188776620705661560	2019741830732340	1452494754364	8192		15154281952019402897256
14	12135636123277455360000	84949452862942187520000	262215386498591285760000	468949431773299276800000	536813469277105692672000	409992848363740561920000	211198322705940041472000	72488018890256606208000	15988801293192490963200	2116079205317511033600	148561096304079459000	4378217629183398000	32331846734015444	13072452822044	16384	2077000418652515559419672

Technical details

Node id: n-a466434b299cb969 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-8c4e3f9383454e05

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:            44,             44,              4;
  3:           792,           1188,            412,              8;
  4:         19800,          39600,          23540,           3740,             16;
  5:        633600,        1584000,        1334520,         417780,          33724,            32;
  6:      24710400,       74131200,       80883000,       38214000,        7055444,        303644,           64;
  7:    1136678400,     3978374400,     5331110400,     3381840000,     1005919992,     116227188,      2733052,         128;
  8:   60243955200,   240975820800,   382961779200,   305469964800,   126603199800,   25228249200,   1889698580,    24597980,       256;
  9: 3614637312000, 16265867904000, 29906008704000, 28763646912000, 15353691091200, 4428802224000, 614264306040, 30505755060, 221382844, 512;

%F Formula

The row polynomials satisfy P(n,x) = (7*n-3+(7*n-5)*x)*P(n-1,x) - (7*x+7*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[(7*n - 3 + (7*n - 5)*x)*P[n - 1, x] - (7*x + 7*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] = (7*n - 3 + (7*n - 5)*x)*v[i - 1] - (7*x + 7*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) = 7*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) = 7*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257617 → 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