A090452: Scaled array A078740 ((3,2)-Stirling2).

A090452 ↗ nonnegative coefficients

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

Recurrence from A090452

\(\left(2 + n\right)\,P_{n}(x) = \left(1 + 3\,x + 2\,x^{2} + 2\,n + 6\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(1 + 2\,x + x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 1 + 3\,x + 2\,x^{2}\)

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	3	2													6
2	1	7	16	15	5											44
3	1	12	51	105	114	63	14									360
4	1	18	118	396	771	910	644	252	42							3152
5	1	25	230	1110	3235	6083	7580	6240	3270	990	132					28896
6	1	33	402	2600	10365	27483	50464	65331	59625	37620	15642	3861	429			273856
7	1	42	651	5390	27825	97188	240751	434493	578690	569415	409332	209209	72072	15015	1430	2661504

All 15 rows b-files: A090442 (row sums), A090452 ?

Technical details

Node id: n-4e0759372c05411f · revision 0 · JSON
Definition source: oeis_catalog:A090452
Fingerprint: f6-2df49debe483a37b

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A090452 triangle 55 terms, 7 rows · tabf · nonn · catalog b-file?

Row sums ? A090442

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  3,    2;
  2: 1,  7,   16,    15,      5;
  3: 1, 12,   51,   105,    114,     63,      14;
  4: 1, 18,  118,   396,    771,    910,     644,     252,       42;
  5: 1, 25,  230,  1110,   3235,   6083,    7580,    6240,     3270,      990,      132;
  6: 1, 33,  402,  2600,  10365,  27483,   50464,   65331,    59625,    37620,    15642,     3861,      429;
  7: 1, 42,  651,  5390,  27825,  97188,  240751,  434493,   578690,   569415,   409332,   209209,    72072,    15015,     1430;
  8: 1, 52,  996, 10220,  65828, 289884,  918484, 2161288,  3849715,  5241060,  5462072,  4328324,  2564562,  1101100,   323752,   58344,    4862;
  9: 1, 63, 1458, 18096, 141552, 762048, 2979312, 8757108, 19793907, 34895685, 48344922, 52723476, 45072846, 29875482, 15049008, 5569200, 1428102, 226746, 16796;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (2*n+1+(6*n+3)*x+(4*n+2)*x^2)*P(n-1,x) - (n-1+(2*n-2)*x+(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x + 2*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (6*n + 3)*x + (4*n + 2)*x^2)*P[n - 1, x] - (n - 1 + (2*n - 2)*x + (n - 1)*x^2)*P[n - 2, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1 + (6*n + 3)*x + (4*n + 2)*x^2)*v[i - 1] - (n - 1 + (2*n - 2)*x + (n - 1)*x^2)*v[i - 2])/(n + 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

A090452 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,  3,    2;
   3: 1,  7,   16,    15,      5;
   4: 1, 12,   51,   105,    114,     63,      14;
   5: 1, 18,  118,   396,    771,    910,     644,     252,       42;
   6: 1, 25,  230,  1110,   3235,   6083,    7580,    6240,     3270,      990,      132;
   7: 1, 33,  402,  2600,  10365,  27483,   50464,   65331,    59625,    37620,    15642,     3861,      429;
   8: 1, 42,  651,  5390,  27825,  97188,  240751,  434493,   578690,   569415,   409332,   209209,    72072,    15015,     1430;
   9: 1, 52,  996, 10220,  65828, 289884,  918484, 2161288,  3849715,  5241060,  5462072,  4328324,  2564562,  1101100,   323752,   58344,    4862;
  10: 1, 63, 1458, 18096, 141552, 762048, 2979312, 8757108, 19793907, 34895685, 48344922, 52723476, 45072846, 29875482, 15049008, 5569200, 1428102, 226746, 16796;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = (2*n-1+(6*n-3)*x+(4*n-2)*x^2)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x + 2*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 3*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (6*n - 3)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x + 2*x^2; for(n = 3, N, my(i = n); v[i] = ((2*n - 1 + (6*n - 3)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n + 1)); 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 ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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