A080248 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(4 + 2\,x\right)\,P_{n-1}(x) + \left(4 + 4\,x\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	6	6	1													13
3	25	36	13	1												75
4	135	250	139	24	1											549
5	909	2041	1517	423	40	1										4931
6	7417	19444	17976	6980	1093	62	1									52973
7	71698	213989	235594	117236	26343	2499	91	1								667451
8	806968	2692926	3428037	2077040	615059	85164	5194	128	1							9710517
9	10427825	38378014	55304019	39428297	14550814	2709503	242986	10002	174	1						161051635
10	152915697	614048886	985950595	807126596	357171201	84575334	10321081	625624	18096	230	1					3012753341
11	2519879761	10946194599	19344249409	17853520511	9213942047	2664452055	419428315	34780361	1478785	31086	297	1				62997957227
12	46276398129	215930716144	415881838876	426656336520	251501579638	86178264068	16816979386	1813893288	105615576	3252326	51117	376	1			1461164925445
13	940296067422	4685259305521	9755989104344	11000915774036	7288284223558	2891828282394	678405926518	91652436250	6967219389	293389382	6726863	80977	468	1		37339898537123
14	21007099850230	111220015237678	248717877665709	305450029913840	224544400720554	101326256511232	27893218724308	4581747881528	438852281359	24141433446	754809055	13197132	124215	574	1	1045204408350861

Technical details

Node id: n-dd053b6394a3f024 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-88b05160e47c12f9

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        2,        1;
  2:        6,        6,        1;
  3:       25,       36,       13,        1;
  4:      135,      250,      139,       24,        1;
  5:      909,     2041,     1517,      423,       40,       1;
  6:     7417,    19444,    17976,     6980,     1093,      62,      1;
  7:    71698,   213989,   235594,   117236,    26343,    2499,     91,     1;
  8:   806968,  2692926,  3428037,  2077040,   615059,   85164,   5194,   128,   1;
  9: 10427825, 38378014, 55304019, 39428297, 14550814, 2709503, 242986, 10002, 174, 1;

%F Formula

The row polynomials satisfy 2*P(n,x) = (4+2*x)*P(n-1,x) + (4+4*x)*d/dx P(n-1,x) + (1+2*x+x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4 + 2*x)*P[n - 1, x] + (4 + 4*x)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*D[P[n - 1, x], {x, 2}])/(2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4 + 2*x)*v[i - 1] + (4 + 4*x)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*deriv(deriv(v[i - 1], x), x))/(2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Stirling-like number triangle defined by sequence A000217.?
  - 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.