A135089: Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.

A135089 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A135089

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	5														10
2	5	10	5													20
3	5	15	15	5												40
4	5	20	30	20	5											80
5	5	25	50	50	25	5										160
6	5	30	75	100	75	30	5									320
7	5	35	105	175	175	105	35	5								640
8	5	40	140	280	350	280	140	40	5							1280
9	5	45	180	420	630	630	420	180	45	5						2560
10	5	50	225	600	1050	1260	1050	600	225	50	5					5120
11	5	55	275	825	1650	2310	2310	1650	825	275	55	5				10240
12	5	60	330	1100	2475	3960	4620	3960	2475	1100	330	60	5			20480
13	5	65	390	1430	3575	6435	8580	8580	6435	3575	1430	390	65	5		40960
14	5	70	455	1820	5005	10010	15015	17160	15015	10010	5005	1820	455	70	5	81920

b-files: A135089, A135089 (reversed triangle) ?

Technical details

Node id: n-f0857c877d3df8d4 · revision 1 · JSON
Definition source: oeis_catalog:A135089
Fingerprint: f6-77a7ed3b1d5f257e

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

A135089 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
A135089 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

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

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 5,  5;
  2: 5, 10,   5;
  3: 5, 15,  15,   5;
  4: 5, 20,  30,  20,   5;
  5: 5, 25,  50,  50,  25,   5;
  6: 5, 30,  75, 100,  75,  30,   5;
  7: 5, 35, 105, 175, 175, 105,  35,   5;
  8: 5, 40, 140, 280, 350, 280, 140,  40,  5;
  9: 5, 45, 180, 420, 630, 630, 420, 180, 45, 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) = 1, P(1,x) = 5 + 5*x.

%t Mathematica

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

A135089 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 5,  5;
  2: 5, 10,   5;
  3: 5, 15,  15,   5;
  4: 5, 20,  30,  20,   5;
  5: 5, 25,  50,  50,  25,   5;
  6: 5, 30,  75, 100,  75,  30,   5;
  7: 5, 35, 105, 175, 175, 105,  35,   5;
  8: 5, 40, 140, 280, 350, 280, 140,  40,  5;
  9: 5, 45, 180, 420, 630, 630, 420, 180, 45, 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) = 1, P(1,x) = 5 + 5*x.

%t Mathematica

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

A135089 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 5,  5;
  2: 5, 10,   5;
  3: 5, 15,  15,   5;
  4: 5, 20,  30,  20,   5;
  5: 5, 25,  50,  50,  25,   5;
  6: 5, 30,  75, 100,  75,  30,   5;
  7: 5, 35, 105, 175, 175, 105,  35,   5;
  8: 5, 40, 140, 280, 350, 280, 140,  40,  5;
  9: 5, 45, 180, 420, 630, 630, 420, 180, 45, 5;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (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) = 5 + 5*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
Table[Reverse[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] = 5 + 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 = Vecrev(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 Iterated real-rootedness preservers, Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Liu–Wang theorem, 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

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

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.