A225467 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(7 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	7	4														11
3	49	72	16													137
4	343	988	528	64												1923
5	2401	12240	11872	3328	256											30097
6	16807	144244	227040	110720	19200	1024										519035
7	117649	1653912	3982576	3011840	884480	104448	4096									9759001
8	823543	18663628	66354288	73155264	32390400	6358016	544768	16384								198306291
9	5764801	208594080	1069968832	1655367168	1037600256	301228032	42319872	2752512	65536							4323661089
10	40353607	2317594084	16883908800	35731851520	30486274560	12283557888	2516828160	265617408	13565952	262144						100539814123
11	282475249	25654949352	262529008336	746440814080	844111720960	453601161216	127155904512	19363921920	1591541760	65536000	1048576					2480798081961
12	1977326743	283334343868	4040554922448	15232491500864	22400332838400	15623678236672	5756237684736	1186360885248	139525816320	9184215040	311427072	4194304				64673993391715
13	13841287201	3124587089520	61741661212192	305579558206208	576137621286656	511440643743744	240938081173504	64547581722624	10186950377472	953024512000	51373932544	1459617792	16777216			1774716400938673
14	96889010407	34425823133524	938623266540960	6052978250766720	14473483522417920	16113447866227712	9514843091353600	3222917684985856	655481391611904	81727855525888	6226672877568	279936237568	6761218048	67108864		51094539079016539
15	678223072849	379071610510392	14217052290648496	118761079830731520	357102034018679040	492957026477819904	359413927296872448	150861491339919360	38455445012807680	6136223354060800	619565047349248	39183439626240	1491611942912	31004295168	268435456	1538944300826771513

Technical details

Node id: n-a13955bc75263637 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-01a7a97b7ee8f4b3

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:        7,          4;
   3:       49,         72,          16;
   4:      343,        988,         528,          64;
   5:     2401,      12240,       11872,        3328,         256;
   6:    16807,     144244,      227040,      110720,       19200,        1024;
   7:   117649,    1653912,     3982576,     3011840,      884480,      104448,       4096;
   8:   823543,   18663628,    66354288,    73155264,    32390400,     6358016,     544768,     16384;
   9:  5764801,  208594080,  1069968832,  1655367168,  1037600256,   301228032,   42319872,   2752512,    65536;
  10: 40353607, 2317594084, 16883908800, 35731851520, 30486274560, 12283557888, 2516828160, 265617408, 13565952, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (7+4*x)*P(n-1,x) + 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[(7 + 4*x)*P[n - 1, x] + 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] = (7 + 4*x)*v[i - 1] + 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n …?
  - 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

Related ?

Same fingerprint ?

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.