A112493 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	8	10	3													21
4	52	106	70	15												243
5	472	1336	1390	630	105											3933
6	5504	19944	28476	20020	6930	945										81819
7	78416	346368	629916	603568	321300	90090	10395									2080053
8	1320064	6878624	15211176	18500300	13362580	5731110	1351350	135135								62490339
9	25637824	153948704	401077056	592017800	541428580	314113800	112882770	22972950	2027025							2166106509
10	564275648	3836381792	11508826736	19991614456	22156780360	16246009780	7879932060	2437835400	436486050	34459425						85092601707
11	13879795712	105387408512	357799940960	715192452832	931957034008	827150364040	506337872040	211073422560	57340483200	9166207050	654729075					3735939709989
12	377332365568	3164688644352	11996202556416	27126097014016	40651166796240	42388469803680	31379569901680	16490614500360	6028570728180	1460045837250	210822762150	13749310575				181287330220467
13	11234698041088	103147816922112	431825383032576	1089930304468096	1847055323153040	2213850662063040	1924239256619680	1221970282893360	562659617520000	183189703296600	40028826187350	5270569053750	316234143225			9634718677393917
14	363581406419456	3626865494340352	16620158924369664	46322220333479168	87600829894098976	118692480772866000	118532552089471520	88332059657361520	49117885782966000	20127206326827600	5907394343951100	1175969367272700	142305364451250	7905853580625		556569415611455931
15	12707452084972544	136854354285477376	681359641827743232	2078462469946206464	4339624960538478208	6559979986353419808	7403380188840075440	6336491361981901120	4132860512223162720	2044084039831033200	754629817074833700	201639322630746000	36862359961927500	4126855569086250	213458046676875	34723276781195740437

Technical details

Node id: n-ded0a7d0241d1354 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-6f934efd2b61885b

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:         2,          1;
   3:         8,         10,           3;
   4:        52,        106,          70,          15;
   5:       472,       1336,        1390,         630,         105;
   6:      5504,      19944,       28476,       20020,        6930,         945;
   7:     78416,     346368,      629916,      603568,      321300,       90090,      10395;
   8:   1320064,    6878624,    15211176,    18500300,    13362580,     5731110,    1351350,     135135;
   9:  25637824,  153948704,   401077056,   592017800,   541428580,   314113800,  112882770,   22972950,   2027025;
  10: 564275648, 3836381792, 11508826736, 19991614456, 22156780360, 16246009780, 7879932060, 2437835400, 436486050, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (n+(n-1)*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[(n + (n - 1)*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] = (n + (n - 1)*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 read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201…?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

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