A048854 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(16 + 8\,x\right)\,P_{n-1}'(x) + \left(16 + 16\,x\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	1														4
3	25	14	1													40
4	331	243	33	1												608
5	5937	5212	1014	60	1											12224
6	133651	133925	33310	2890	95	1										303872
7	3609673	4016106	1201215	135980	6615	138	1									8969728
8	113525595	137718679	47690391	6610275	423185	13125	189	1								305981440
9	4070168161	5315202488	2076967228	338703176	26762470	1097096	23548	248	1							11828924416
10	163716695587	227978803017	98689231404	18424216020	1725531066	87667902	2493372	39204	315	1						510624677888
11	7296866339961	10756319076550	5087463271245	1065932745480	115081782690	6957778212	245848050	5130120	61605	390	1					24328872034304
12	356790528924523	553543569107771	283008138855805	65555072795685	7994913488430	559322163246	23573240922	612057930	9764535	92455	473	1				1267475740491776
13	18983329135883665	30851577664267284	16905652079725266	4278676049616100	580421065831695	46035256208040	2250159890748	69790050792	1386784575	17457220	133650	564	1			71648012605849600
14	1091556096587136051	1851065317308386413	1079629162963451178	295748980258846686	44087592584500465	3902928093980487	216751369436316	7807951009716	185521630437	2910908715	29644186	187278	663	1		4366214825579118592
15	67439846581526028457	118941443121489543842	73416690347111970859	21600212321658186580	3504026615637989441	342009232499508574	21244908907721499	870747995416728	24053125571475	451600090942	5735346617	48216532	255619	770	1	285266368387335847936

Technical details

Node id: n-97d149f0803a3506 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-8b727355ed6cd2e7

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,            1;
   3:           25,           14,           1;
   4:          331,          243,          33,           1;
   5:         5937,         5212,        1014,          60,          1;
   6:       133651,       133925,       33310,        2890,         95,        1;
   7:      3609673,      4016106,     1201215,      135980,       6615,      138,       1;
   8:    113525595,    137718679,    47690391,     6610275,     423185,    13125,     189,     1;
   9:   4070168161,   5315202488,  2076967228,   338703176,   26762470,  1097096,   23548,   248,   1;
  10: 163716695587, 227978803017, 98689231404, 18424216020, 1725531066, 87667902, 2493372, 39204, 315, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (16+8*x)*d/dx P(n-1,x) + (16+16*x)*d^2/dx^2 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 + x)*P[n - 1, x] + (16 + 8*x)*D[P[n - 1, x], x] + (16 + 16*x)*D[P[n - 1, x], {x, 2}]];
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 + x)*v[i - 1] + (16 + 8*x)*deriv(v[i - 1], x) + (16 + 16*x)*deriv(deriv(v[i - 1], x), 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)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.