A096940: Pascal (1,5) triangle.

A096940 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A096940

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

\(P_{0}(x) = 5\)

\(P_{1}(x) = 1 + 5\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	5															5
1	1	5														6
2	1	6	5													12
3	1	7	11	5												24
4	1	8	18	16	5											48
5	1	9	26	34	21	5										96
6	1	10	35	60	55	26	5									192
7	1	11	45	95	115	81	31	5								384
8	1	12	56	140	210	196	112	36	5							768
9	1	13	68	196	350	406	308	148	41	5						1536
10	1	14	81	264	546	756	714	456	189	46	5					3072
11	1	15	95	345	810	1302	1470	1170	645	235	51	5				6144
12	1	16	110	440	1155	2112	2772	2640	1815	880	286	56	5			12288
13	1	17	126	550	1595	3267	4884	5412	4455	2695	1166	342	61	5		24576
14	1	18	143	676	2145	4862	8151	10296	9867	7150	3861	1508	403	66	5	49152

b-files: A091629 (row sums), A096940 ?

Technical details

Node id: n-6fc8b9c5d9c18144 · revision 1 · JSON
Definition source: oeis_catalog:A096940
Fingerprint: f6-5bcf47c8067a6574

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 ?

A096940 triangle 72 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A003945, A167881, A111286, A042950, A058764, A007283, A091629, A082505

Submission text ?

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

%e Example

Triangle begins:
  0: 5;
  1: 1,  5;
  2: 1,  6,  5;
  3: 1,  7, 11,   5;
  4: 1,  8, 18,  16,   5;
  5: 1,  9, 26,  34,  21,   5;
  6: 1, 10, 35,  60,  55,  26,   5;
  7: 1, 11, 45,  95, 115,  81,  31,   5;
  8: 1, 12, 56, 140, 210, 196, 112,  36,  5;
  9: 1, 13, 68, 196, 350, 406, 308, 148, 41, 5;

%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) = 5, P(1,x) = 1 + 5*x.

%t Mathematica

P[0, x_] := 5;
P[1, x_] := 1 + 5*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] = 5; v[2] = 1 + 5*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

A096940 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 5;
  1: 1,  5;
  2: 1,  6,  5;
  3: 1,  7, 11,   5;
  4: 1,  8, 18,  16,   5;
  5: 1,  9, 26,  34,  21,   5;
  6: 1, 10, 35,  60,  55,  26,   5;
  7: 1, 11, 45,  95, 115,  81,  31,   5;
  8: 1, 12, 56, 140, 210, 196, 112,  36,  5;
  9: 1, 13, 68, 196, 350, 406, 308, 148, 41, 5;

%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) = 5, P(1,x) = 1 + 5*x.

%t Mathematica

P[0, x_] := 5;
P[1, x_] := 1 + 5*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] = 5; v[2] = 1 + 5*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 ?

A093562: (5,1) 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.