A386789: Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).

A386789 ↗ nonnegative coefficients

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

Recurrence from A386789

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(1 + 2\,x - 3\,n - 6\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,n - n^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	3	6						9
3	0	4	20	20					44
4	0	5	45	105	70				225
5	0	6	84	336	504	252			1182
6	0	7	140	840	2100	2310	924		6321
7	0	8	216	1800	6600	11880	10296	3432	34232

All 15 rows b-files: A176479 (row sums), A386789 ?

Technical details

Node id: n-2fa133578d3075fe · revision 1 · JSON
Definition source: oeis_catalog:A386789
Fingerprint: f6-1e5fbd8fe1b88630

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): no

Matches ?

A386789 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A176479

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0,  2;
  2: 0,  3,   6;
  3: 0,  4,  20,   20;
  4: 0,  5,  45,  105,    70;
  5: 0,  6,  84,  336,   504,    252;
  6: 0,  7, 140,  840,  2100,   2310,    924;
  7: 0,  8, 216, 1800,  6600,  11880,  10296,   3432;
  8: 0,  9, 315, 3465, 17325,  45045,  63063,  45045,  12870;
  9: 0, 10, 440, 6160, 40040, 140140, 280280, 320320, 194480, 48620;

%F Formula

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

%t Mathematica

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

A386789 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0,  2;
  2: 0,  3,   6;
  3: 0,  4,  20,   20;
  4: 0,  5,  45,  105,    70;
  5: 0,  6,  84,  336,   504,    252;
  6: 0,  7, 140,  840,  2100,   2310,    924;
  7: 0,  8, 216, 1800,  6600,  11880,  10296,   3432;
  8: 0,  9, 315, 3465, 17325,  45045,  63063,  45045,  12870;
  9: 0, 10, 440, 6160, 40040, 140140, 280280, 320320, 194480, 48620;

%F Formula

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

%t Mathematica

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

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] 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

Used in 12

4 more

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.