A281260: Triangular array of generalized Narayana numbers T(n,k) = 2binomial(n+1,k) binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.

A281260 ↗ nonnegative coefficients

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

Recurrence from A281260

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	2	3													5
3	0	2	8	4												14
4	0	2	15	20	5											42
5	0	2	24	60	40	6										132
6	0	2	35	140	175	70	7									429
7	0	2	48	280	560	420	112	8								1430
8	0	2	63	504	1470	1764	882	168	9							4862
9	0	2	80	840	3360	5880	4704	1680	240	10						16796
10	0	2	99	1320	6930	16632	19404	11088	2970	330	11					58786
11	0	2	120	1980	13200	41580	66528	55440	23760	4950	440	12				208012
12	0	2	143	2860	23595	94380	198198	226512	141570	47190	7865	572	13			742900
13	0	2	168	4004	40040	198198	528528	792792	679536	330330	88088	12012	728	14		2674440
14	0	2	195	5460	65065	390390	1288287	2453880	2760615	1840410	715715	156156	17745	910	15	9694845

b-files: A108838 (reversed triangle), A281260 ?

Technical details

Node id: n-8a65e165240c40e0 · revision 1 · JSON
Definition source: oeis_catalog:A281260
Fingerprint: f6-4e2cf4a8bda0ea40

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 ?

A108838 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A281260 triangle 72 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A000108, A120588, A230585

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 2,  3;
  3: 0, 2,  8,   4;
  4: 0, 2, 15,  20,    5;
  5: 0, 2, 24,  60,   40,    6;
  6: 0, 2, 35, 140,  175,   70,    7;
  7: 0, 2, 48, 280,  560,  420,  112,    8;
  8: 0, 2, 63, 504, 1470, 1764,  882,  168,   9;
  9: 0, 2, 80, 840, 3360, 5880, 4704, 1680, 240, 10;

%F Formula

The row polynomials satisfy (2*n^2+3*n-2)*P(n,x) = (4*n^2-4+(4*n^2+2)*x)*P(n-1,x) - (2*n^2-3*n-2-(4*n^2-6*n-4)*x+(2*n^2-3*n-2)*x^2)*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[((4*n^2 - 4 + (4*n^2 + 2)*x)*P[n - 1, x] - (2*n^2 - 3*n - 2 - (4*n^2 - 6*n - 4)*x + (2*n^2 - 3*n - 2)*x^2)*P[n - 2, x])/(2*n^2 + 3*n - 2)]];
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] = ((4*n^2 - 4 + (4*n^2 + 2)*x)*v[i - 1] - (2*n^2 - 3*n - 2 - (4*n^2 - 6*n - 4)*x + (2*n^2 - 3*n - 2)*x^2)*v[i - 2])/(2*n^2 + 3*n - 2)); 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

A108838 reversed triangle, offset 2, from P₁, low zeros dropped

%e Example

Triangle begins:
   2:  2;
   3:  3,   2;
   4:  4,   8,    2;
   5:  5,  20,   15,     2;
   6:  6,  40,   60,    24,     2;
   7:  7,  70,  175,   140,    35,     2;
   8:  8, 112,  420,   560,   280,    48,    2;
   9:  9, 168,  882,  1764,  1470,   504,   63,    2;
  10: 10, 240, 1680,  4704,  5880,  3360,  840,   80,  2;
  11: 11, 330, 2970, 11088, 19404, 16632, 6930, 1320, 99, 2;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (2*n^2-n-3)*P(n,x) = (4*n^2-8*n+(4*n^2-8*n+6)*x)*P(n-1,x) - (2*n^2-7*n+3-(4*n^2-14*n+6)*x+(2*n^2-7*n+3)*x^2)*P(n-2,x) for n >= 4, with P(2,x) = 2*x, P(3,x) = 2*x + 3*x^2.

%t Mathematica

P[2, x_] := 2*x;
P[3, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 8*n + (4*n^2 - 8*n + 6)*x)*P[n - 1, x] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*P[n - 2, x])/(2*n^2 - n - 3)]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n - 1]], {n, 2, 11}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N - 1)); v[1] = 2*x; v[2] = 2*x + 3*x^2; for(n = 4, N, my(i = n - 1); v[i] = ((4*n^2 - 8*n + (4*n^2 - 8*n + 6)*x)*v[i - 1] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*v[i - 2])/(2*n^2 - n - 3)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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(11)

polytool c48629374597

A281260 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 0, 2;
   3: 0, 2,  3;
   4: 0, 2,  8,   4;
   5: 0, 2, 15,  20,    5;
   6: 0, 2, 24,  60,   40,    6;
   7: 0, 2, 35, 140,  175,   70,    7;
   8: 0, 2, 48, 280,  560,  420,  112,    8;
   9: 0, 2, 63, 504, 1470, 1764,  882,  168,   9;
  10: 0, 2, 80, 840, 3360, 5880, 4704, 1680, 240, 10;

%F Formula

The row polynomials satisfy (2*n^2-n-3)*P(n,x) = (4*n^2-8*n+(4*n^2-8*n+6)*x)*P(n-1,x) - (2*n^2-7*n+3-(4*n^2-14*n+6)*x+(2*n^2-7*n+3)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 8*n + (4*n^2 - 8*n + 6)*x)*P[n - 1, x] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*P[n - 2, x])/(2*n^2 - n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 8*n + (4*n^2 - 8*n + 6)*x)*v[i - 1] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*v[i - 2])/(2*n^2 - n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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
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

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 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.