A360282: Triangle read by rows. T(n, k) = (1/2) * binomial(2(n - k + 1), n - k + 1) binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.

A360282 ↗ nonnegative coefficients

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

Recurrence from A360282

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(6 + 4\,x - 16\,n - 8\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x^{2} + 5\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	1							1
2	0	3	2						5
3	0	10	12	3					25
4	0	35	60	30	4				129
5	0	126	280	210	60	5			681
6	0	462	1260	1260	560	105	6		3653
7	0	1716	5544	6930	4200	1260	168	7	19825

All 15 rows b-files: A360282 ?

Technical details

Node id: n-0a5e54eacdb24124 · revision 1 · JSON
Definition source: oeis_catalog:A360282
Fingerprint: f6-2c1fea4a558d4e6c

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 ?

A360282 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A002002

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0,     1;
  2: 0,     3,      2;
  3: 0,    10,     12,      3;
  4: 0,    35,     60,     30,      4;
  5: 0,   126,    280,    210,     60,     5;
  6: 0,   462,   1260,   1260,    560,   105,     6;
  7: 0,  1716,   5544,   6930,   4200,  1260,   168,    7;
  8: 0,  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;
  9: 0, 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;

%F Formula

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = (8*n^2-16*n+6+(4*n^2-8*n+4)*x)*P(n-1,x) - (2*n^2-5*n+2)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] - (2*n^2 - 5*n + 2)*x^2*P[n - 2, x])/(2*n^2 - 3*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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*v[i - 1] - (2*n^2 - 5*n + 2)*x^2*v[i - 2])/(2*n^2 - 3*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

A360282 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0,     1;
  2: 0,     3,      2;
  3: 0,    10,     12,      3;
  4: 0,    35,     60,     30,      4;
  5: 0,   126,    280,    210,     60,     5;
  6: 0,   462,   1260,   1260,    560,   105,     6;
  7: 0,  1716,   5544,   6930,   4200,  1260,   168,    7;
  8: 0,  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;
  9: 0, 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;

%F Formula

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = (8*n^2-16*n+6+(4*n^2-8*n+4)*x)*P(n-1,x) - (2*n^2-5*n+2)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] - (2*n^2 - 5*n + 2)*x^2*P[n - 2, x])/(2*n^2 - 3*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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*v[i - 1] - (2*n^2 - 5*n + 2)*x^2*v[i - 2])/(2*n^2 - 3*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 Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

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 for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 11

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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.