A269945 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	2	3	1													6
4	7	14	8	1												30
5	37	89	69	18	1											214
6	264	737	716	277	35	1										2030
7	2433	7696	8892	4486	917	61	1									24486
8	27913	98093	130450	80648	22850	2569	98	1								362622
9	386906	1490687	2234263	1628110	587643	96154	6286	148	1							6430198
10	6346119	26542518	44135368	36892447	16133551	3508739	342260	13806	213	1						133915022
11	121159373	545598045	994208675	934974046	479323869	130529777	17440633	1058192	27804	295	1					3224320710
12	2655174768	12787451789	25293138538	26375613007	15480538774	5068623913	873685694	73707953	2910677	52184	396	1				88610897694
13	66028903633	338242587616	720653804872	823876832540	543843419070	208023750342	44265416226	4912121832	270921750	7265115	92411	518	1			2750125115926
14	1845579100993	10009697903337	22825785128916	28347512871072	20758246208900	9080287515822	2308018855416	322905206106	23641915167	884276250	16718416	155883	663	1		95522575856942
15	57506847262162	328904762907595	798450271120797	1069137558005276	859091202296842	422736984734180	125185306171212	21353976870756	1996428615651	99346213837	2608932755	35919286	252343	833	1	3684465329303526

Technical details

Node id: n-63f3fd588595ef25 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-34367c82c455b0b6

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:       1,        1;
   3:       2,        3,        1;
   4:       7,       14,        8,        1;
   5:      37,       89,       69,       18,        1;
   6:     264,      737,      716,      277,       35,       1;
   7:    2433,     7696,     8892,     4486,      917,      61,      1;
   8:   27913,    98093,   130450,    80648,    22850,    2569,     98,     1;
   9:  386906,  1490687,  2234263,  1628110,   587643,   96154,   6286,   148,   1;
  10: 6346119, 26542518, 44135368, 36892447, 16133551, 3508739, 342260, 13806, 213, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) + (1+2*x+x^2)*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[(1 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*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] = (1 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*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. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) =…?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.