A093644: (9,1) Pascal triangle.

A093644 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A093644

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

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

\(P_{1}(x) = 9 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	9	1														10
2	9	10	1													20
3	9	19	11	1												40
4	9	28	30	12	1											80
5	9	37	58	42	13	1										160
6	9	46	95	100	55	14	1									320
7	9	55	141	195	155	69	15	1								640
8	9	64	196	336	350	224	84	16	1							1280
9	9	73	260	532	686	574	308	100	17	1						2560
10	9	82	333	792	1218	1260	882	408	117	18	1					5120
11	9	91	415	1125	2010	2478	2142	1290	525	135	19	1				10240
12	9	100	506	1540	3135	4488	4620	3432	1815	660	154	20	1			20480
13	9	109	606	2046	4675	7623	9108	8052	5247	2475	814	174	21	1		40960
14	9	118	715	2652	6721	12298	16731	17160	13299	7722	3289	988	195	22	1	81920

b-files: A093644 ?

Technical details

Node id: n-2bc3c94571dff60c · revision 1 · JSON
Definition source: oeis_catalog:A093644
Fingerprint: f6-e75dc09dd574571b

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 ?

A093644 triangle 69 terms, 11 rows · tabl · nonn · catalog b-file?

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

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 9,  1;
  2: 9, 10,   1;
  3: 9, 19,  11,   1;
  4: 9, 28,  30,  12,   1;
  5: 9, 37,  58,  42,  13,   1;
  6: 9, 46,  95, 100,  55,  14,   1;
  7: 9, 55, 141, 195, 155,  69,  15,   1;
  8: 9, 64, 196, 336, 350, 224,  84,  16,  1;
  9: 9, 73, 260, 532, 686, 574, 308, 100, 17, 1;

%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) = 1, P(1,x) = 9 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 9 + 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] = 1; v[2] = 9 + 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

A093644 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 9,  1;
  2: 9, 10,   1;
  3: 9, 19,  11,   1;
  4: 9, 28,  30,  12,   1;
  5: 9, 37,  58,  42,  13,   1;
  6: 9, 46,  95, 100,  55,  14,   1;
  7: 9, 55, 141, 195, 155,  69,  15,   1;
  8: 9, 64, 196, 336, 350, 224,  84,  16,  1;
  9: 9, 73, 260, 532, 686, 574, 308, 100, 17, 1;

%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) = 1, P(1,x) = 9 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 9 + 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] = 1; v[2] = 9 + 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 Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem
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

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
Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.