A134239: A127899(unsigned) * A007318.

A134239 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A134239

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

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

\(P_{1}(x) = 4 + 2\,x\)

\(P_{2}(x) = 6 + 9\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	6	9	3													18
3	8	20	16	4												48
4	10	35	45	25	5											120
5	12	54	96	84	36	6										288
6	14	77	175	210	140	49	7									672
7	16	104	288	440	400	216	64	8								1536
8	18	135	441	819	945	693	315	81	9							3456
9	20	170	640	1400	1960	1820	1120	440	100	10						7680
10	22	209	891	2244	3696	4158	3234	1716	594	121	11					16896
11	24	252	1200	3420	6480	8568	8064	5400	2520	780	144	12				36864
12	26	299	1573	5005	10725	16302	18018	14586	8580	3575	1001	169	13			79872
13	28	350	2016	7084	16940	29106	36960	35112	24948	13090	4928	1260	196	14		172032
14	30	405	2535	9750	25740	49335	70785	77220	64350	40755	19305	6630	1560	225	15	368640

b-files: A128543 (row sums), A134239 ?

Technical details

Node id: n-e917c954825de4ec · revision 1 · JSON
Definition source: oeis_catalog:A134239
Fingerprint: f6-efce8cd29f4f1fd0

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): yes

Matches ?

A134239 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A128543

Submission text ?

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

%e Example

Triangle begins:
  0:  1;
  1:  4,   2;
  2:  6,   9,   3;
  3:  8,  20,  16,    4;
  4: 10,  35,  45,   25,    5;
  5: 12,  54,  96,   84,   36,    6;
  6: 14,  77, 175,  210,  140,   49,    7;
  7: 16, 104, 288,  440,  400,  216,   64,   8;
  8: 18, 135, 441,  819,  945,  693,  315,  81,   9;
  9: 20, 170, 640, 1400, 1960, 1820, 1120, 440, 100, 10;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 6 + 9*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 6 + 9*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*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] = 4 + 2*x; v[3] = 6 + 9*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*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

A134239 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:  1;
  1:  4,   2;
  2:  6,   9,   3;
  3:  8,  20,  16,    4;
  4: 10,  35,  45,   25,    5;
  5: 12,  54,  96,   84,   36,    6;
  6: 14,  77, 175,  210,  140,   49,    7;
  7: 16, 104, 288,  440,  400,  216,   64,   8;
  8: 18, 135, 441,  819,  945,  693,  315,  81,   9;
  9: 20, 170, 640, 1400, 1960, 1820, 1120, 440, 100, 10;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 6 + 9*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 6 + 9*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n - 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; v[2] = 4 + 2*x; v[3] = 6 + 9*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n - 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 Row-hypergeometric rows (multiplier sequences), 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 Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

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

Used in 5

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.