A019538 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−2, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	2														5
3	13	18	6													37
4	75	158	108	24												365
5	541	1530	1590	720	120											4501
6	4683	16622	23220	15960	5400	720										66605
7	47293	201978	354606	327600	168000	45360	5040									1149877
8	545835	2724878	5762988	6693624	4611600	1884960	423360	40320								22687565
9	7087261	40492890	100203270	140268240	121491720	66679200	22649760	4354560	362880							503589781
10	102247563	657944942	1864917540	3055965720	3190509000	2200973040	1003363200	291513600	48988800	3628800						12420052205
11	1622632573	11611834938	37093887006	69655330800	85147194000	70800927120	40558628880	15807052800	4011638400	598752000	39916800					336947795317
12	28091567595	221291822798	786612472668	1665417069624	2333493835200	2272003398720	1568636435520	768024512640	261355248000	58877280000	7903526400	479001600				9972186170765
13	526858348381	4530383962650	17735890140150	41799405407760	66049526863320	73720661414400	59597732113920	35163363755520	15028603349760	4537941408000	919004486400	112086374400	6227020800			319727684645461
14	10641342970443	99179581033262	424010595595860	1100932640360280	1936946544132600	2438441852006160	2259884096870400	1561160240640000	803925509587200	304804365465600	82710403776000	15212611814400	1699976678400	87178291200		11039636939221805
15	230283190977853	2312402554523898	10719229945624206	30402806419767600	58943279883852000	82632241763351280	86382902367281520	68409726703718400	41242909502995200	18837594135360000	6416968804646400	1580978310912000	266329679616000	27461161728000	1307674368000	408406422098722357

Technical details

Node id: n-9cb46fc45f534e25 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d4e4e266d9f48188

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         1;
   2:         3,         2;
   3:        13,        18,          6;
   4:        75,       158,        108,         24;
   5:       541,      1530,       1590,        720,        120;
   6:      4683,     16622,      23220,      15960,       5400,        720;
   7:     47293,    201978,     354606,     327600,     168000,      45360,       5040;
   8:    545835,   2724878,    5762988,    6693624,    4611600,    1884960,     423360,     40320;
   9:   7087261,  40492890,  100203270,  140268240,  121491720,   66679200,   22649760,   4354560,   362880;
  10: 102247563, 657944942, 1864917540, 3055965720, 3190509000, 2200973040, 1003363200, 291513600, 48988800, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) + (2+3*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] + (2 + 3*x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3 + 2*x)*v[i - 1] + (2 + 3*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (-2, -1)
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

Related ?

Same fingerprint ?

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741