A225469 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + x\right)\,P_{n-1}(x) + \left(4 + 4\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	1														5
3	20	12	1													33
4	128	124	24	1												277
5	1008	1312	424	40	1											2785
6	9280	14896	6880	1080	60	1										32197
7	96704	183488	110416	25120	2300	84	1									418113
8	1120768	2447936	1809920	549136	72800	4340	112	1								6005013
9	14274816	35183616	30756608	11760896	2091936	179648	7504	144	1							94255169
10	197833728	541796608	545393664	252401920	57192576	6583584	393792	12144	180	1						1601608197
11	2958521344	8895355904	10115343616	5498905600	1527925120	224649600	17949792	788160	18660	220	1					29239458017
12	47415508992	155044117504	196266346496	122544635136	40550400000	7350310528	749312256	43768032	1467840	27500	264	1				569965894549
13	809838505984	2857899220992	3980775897088	2805786908672	1080558845696	234941346816	29556558592	2196860160	97600272	2578400	39160	312	1			11801654362145
14	14670950907904	55519239450624	84296652890112	66162307966976	29115790758912	7428508575488	1124037071872	102979292416	5803292352	202302672	4315168	54184	364	1		258426476879045
15	280760761434112	1133198089633792	1861026769735680	1608746232504320	797048294654976	234376886295552	41784966775552	4605079784448	319180713280	14068005952	394554160	6933472	73164	420	1	5961880731098881

Technical details

Node id: n-c4be8fdb39895af5 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-3327631615903c42

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:         4,         1;
   3:        20,        12,         1;
   4:       128,       124,        24,         1;
   5:      1008,      1312,       424,        40,        1;
   6:      9280,     14896,      6880,      1080,       60,       1;
   7:     96704,    183488,    110416,     25120,     2300,      84,      1;
   8:   1120768,   2447936,   1809920,    549136,    72800,    4340,    112,     1;
   9:  14274816,  35183616,  30756608,  11760896,  2091936,  179648,   7504,   144,   1;
  10: 197833728, 541796608, 545393664, 252401920, 57192576, 6583584, 393792, 12144, 180, 1;

%F Formula

The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (4+4*x)*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[(4 + x)*P[n - 1, x] + (4 + 4*x)*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] = (4 + x)*v[i - 1] + (4 + 4*x)*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 (−∞, -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, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.