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	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

All 15 rows

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