A080248: Stirling-like number triangle defined by sequence A000217.

A080248 ↗ nonnegative coefficients

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

Recurrence from A080248

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	1	4	1													6
3	1	13	10	1												25
4	1	40	73	20	1											135
5	1	121	478	273	35	1										909
6	1	364	2989	3208	798	56	1									7417
7	1	1093	18298	35069	15178	1974	84	1								71698
8	1	3280	110881	368988	262739	56632	4326	120	1							806968
9	1	9841	668566	3800761	4310073	1452011	177760	8646	165	1						10427825
10	1	29524	4021237	38676176	68451856	34802304	6429291	489016	16071	220	1					152915697
11	1	88573	24156946	390782997	1065454016	799300240	214822452	24033867	1212211	28171	286	1				2519879761
12	1	265720	145030249	3931986916	16372593237	17850759056	6814328896	1080041664	78583362	2761616	47047	364	1			46276398129
13	1	797161	870447214	39464899409	249520885471	391238533413	208651968144	45695828800	4616292954	230472242	5866718	75439	455	1		940296067422
14	1	2391484	5223480445	395519441304	3782278181474	8465530087144	6233493641445	1853701804944	253429011730	17292266264	617675630	11750960	116844	560	1	21007099850230

b-files: A080248 ?

Technical details

Node id: n-15df20466f2d1f05 · revision 0 · JSON
Definition source: oeis_catalog:A080248
Fingerprint: f6-8583efd7a1bc8a2a

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 ?

A080248 triangle 52 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A124373

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    4,      1;
  3: 1,   13,     10,       1;
  4: 1,   40,     73,      20,       1;
  5: 1,  121,    478,     273,      35,       1;
  6: 1,  364,   2989,    3208,     798,      56,      1;
  7: 1, 1093,  18298,   35069,   15178,    1974,     84,    1;
  8: 1, 3280, 110881,  368988,  262739,   56632,   4326,  120,   1;
  9: 1, 9841, 668566, 3800761, 4310073, 1452011, 177760, 8646, 165, 1;

%F Formula

The row polynomials satisfy 2*P(n,x) = (2+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,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[((2 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], 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] = ((2 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], 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

A080248 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    4,      1;
  3: 1,   13,     10,       1;
  4: 1,   40,     73,      20,       1;
  5: 1,  121,    478,     273,      35,       1;
  6: 1,  364,   2989,    3208,     798,      56,      1;
  7: 1, 1093,  18298,   35069,   15178,    1974,     84,    1;
  8: 1, 3280, 110881,  368988,  262739,   56632,   4326,  120,   1;
  9: 1, 9841, 668566, 3800761, 4310073, 1452011, 177760, 8646, 165, 1;

%F Formula

The row polynomials satisfy 2*P(n,x) = (2+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,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[((2 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}])/(2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = ((2 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x))/(2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 (−∞, 0]
by Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers

Proof chains ? JSON

Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Used in 7

Related ?

Same fingerprint ?

A080248 → reverse

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.