A095666: Pascal (1,4) triangle.

A095666 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A095666

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

\(P_{0}(x) = 4\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	4															4
1	1	4														5
2	1	5	4													10
3	1	6	9	4												20
4	1	7	15	13	4											40
5	1	8	22	28	17	4										80
6	1	9	30	50	45	21	4									160
7	1	10	39	80	95	66	25	4								320
8	1	11	49	119	175	161	91	29	4							640
9	1	12	60	168	294	336	252	120	33	4						1280
10	1	13	72	228	462	630	588	372	153	37	4					2560
11	1	14	85	300	690	1092	1218	960	525	190	41	4				5120
12	1	15	99	385	990	1782	2310	2178	1485	715	231	45	4			10240
13	1	16	114	484	1375	2772	4092	4488	3663	2200	946	276	49	4		20480
14	1	17	130	598	1859	4147	6864	8580	8151	5863	3146	1222	325	53	4	40960

b-files: A020714 (row sums), A095666 ?

Technical details

Node id: n-d4825e99260ce554 · revision 1 · JSON
Definition source: oeis_catalog:A095666
Fingerprint: f6-9b47636dd6910c6a

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 ?

A095666 triangle 72 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A146523, A323831, A084215, A020714, A287798, A257113

Submission text ?

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

%e Example

Triangle begins:
  0: 4;
  1: 1,  4;
  2: 1,  5,  4;
  3: 1,  6,  9,   4;
  4: 1,  7, 15,  13,   4;
  5: 1,  8, 22,  28,  17,   4;
  6: 1,  9, 30,  50,  45,  21,   4;
  7: 1, 10, 39,  80,  95,  66,  25,   4;
  8: 1, 11, 49, 119, 175, 161,  91,  29,  4;
  9: 1, 12, 60, 168, 294, 336, 252, 120, 33, 4;

%F Formula

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

%t Mathematica

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

%o PARI/GP

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

A095666 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 4;
  1: 1,  4;
  2: 1,  5,  4;
  3: 1,  6,  9,   4;
  4: 1,  7, 15,  13,   4;
  5: 1,  8, 22,  28,  17,   4;
  6: 1,  9, 30,  50,  45,  21,   4;
  7: 1, 10, 39,  80,  95,  66,  25,   4;
  8: 1, 11, 49, 119, 175, 161,  91,  29,  4;
  9: 1, 12, 60, 168, 294, 336, 252, 120, 33, 4;

%F Formula

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

%t Mathematica

P[0, x_] := 4;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
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] = 4; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (n - 1)*x)*v[i - 1])/(n - 1)); 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 Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem, Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem, Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Iterated real-rootedness preservers

Proof chains ? JSON

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
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

Related ?

Same fingerprint ?

A093561: (4,1) Pascal triangle.

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
[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.