A219512 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	28	42	15													85
3	280	630	456	105												1471
4	3640	10920	11946	5610	945											33061
5	58240	218400	320124	228228	78660	10395										914047
6	1106560	4979520	9154908	8773632	4603320	1246770	135135									29999845
7	24344320	127807680	282688560	340642260	240785136	99451530	22110480	2027025								1139856991
8	608608000	3651648000	9441413520	13713278040	12209975316	6804733320	2309678190	434475090	34459425							49208268901
9	17041024000	115026912000	340419599520	578867289000	622158080544	437338298628	200517053220	57627218880	9378690300	654729075						2379028895167
10	528271744000	3962038080000	13207948834080	25740032878560	32427684977688	27546905858472	15946383796200	6195462530400	1541135911140	220735226250	13749310575					127330349147365
11	17961239296000	148180224192000	549439581250560	1207387433091840	1745031535687440	1739263134248640	1217812110330816	597829850818200	201151559480040	44046778434750	5627628883800	316234143225				7474047309857311
12	664565853952000	5981092685568000	24416693190489600	59732118487641600	97429825969716240	111497429437954560	91669332581963136	54469258834470480	23170460898648960	6865553461614120	1341208997600850	154540438702650	7905853580625			477399986691902821
13	26582634158080000	259180683041280000	1155115863290188800	3113506032090854400	5657830000464882240	7312453939268478000	6907669908677201664	4822573516742309328	2484471031459822080	931116445168155480	246202873776449100	43376733652801500	4548496183931700	213458046676875		32964841616021111167
14	1143053268797440000	12002059322373120000	57986158399590220800	170744875815241728000	342082105269673728960	492870789709867689120	526144582373990147568	422288763315102395712	255746559187074599280	116129974205351237040	38851537525171951140	9266547437736019200	1485682124938729200	142858626918257250	6190283353629375	2446891736865180892645

Technical details

Node id: n-b94ad91373f0cf84 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-678ce348735f9478

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,            3;
  2:          28,           42,           15;
  3:         280,          630,          456,          105;
  4:        3640,        10920,        11946,         5610,          945;
  5:       58240,       218400,       320124,       228228,        78660,        10395;
  6:     1106560,      4979520,      9154908,      8773632,      4603320,      1246770,       135135;
  7:    24344320,    127807680,    282688560,    340642260,    240785136,     99451530,     22110480,     2027025;
  8:   608608000,   3651648000,   9441413520,  13713278040,  12209975316,   6804733320,   2309678190,   434475090,   34459425;
  9: 17041024000, 115026912000, 340419599520, 578867289000, 622158080544, 437338298628, 200517053220, 57627218880, 9378690300, 654729075;

%F Formula

The row polynomials satisfy P(n,x) = (3*n+1+3*n*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[(3*n + 1 + 3*n*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] = (3*n + 1 + 3*n*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 of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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