A316649: Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.

A316649 ↗ nonnegative coefficients

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

Recurrence from A316649

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2} - 5\,n - 15\,n x - 10\,n x^{2} + 2\,n^{2} + 6\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-2 - 4\,x - 2\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x)\)

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

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

\(P_{2}(x) = x + 7\,x^{2} + 12\,x^{3} + 6\,x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	0	1	2																											3
2	0	1	7	12	6																									26
3	0	1	14	55	92	70	20																							252
4	0	1	23	153	471	780	720	350	70																					2568
5	0	1	34	336	1584	4251	7002	7238	4592	1638	252																			26928
6	0	1	47	640	4210	16175	39733	65226	72660	54390	26250	7392	924																	287648
7	0	1	62	1107	9596	49225	164898	380731	623576	732618	614700	360162	140184	32604	3432															3112896
8	0	1	79	1785	19607	128345	555303	1685257	3716695	6077196	7430220	6780312	4557168	2192190	714714	141570	12870													34013312
9	0	1	98	2728	36896	298480	1607648	6139504	17290304	36808723	60110030	75805972	73788176	54980926	30802772	12563980	3523520	607750	48620											374416128
10	0	1	119	3996	65094	635376	4147632	19331088	67077552	178145163	368750757	601585512	777716478	796840902	643620978	404988012	194450256	68831334	16934346	2586584	184756									4145895936
11	0	1	142	5655	109020	1260150	9767988	54309696	226265280	727595595	1843090050	3728940249	6078578508	8019341460	8566871040	7384425750	5094112608	2773012086	1164639060	364095290	79781000	10934196	705432							46127840256
12	0	1	167	7777	174911	2358400	21359228	139183506	682669746	2599775343	7857505425	19143223539	38003358693	61908797418	83068499514	91872151800	83553126228	62122226166	37384516026	17927890838	6690476650	1872686816	369881512	45970652	2704156					515268544512
13	0	1	194	10440	270672	4204695	43910658	330629222	1879818160	8333889201	29481516570	84628436244	199549724256	389866749975	634756203018	863994693690	983962038048	936076904202	740752220340	484042358096	258262169440	110660181786	37153394124	9413583060	1692385632	192411100	10400600			5775088103424
14	0	1	223	13728	406146	7193355	85680153	736815794	4795434644	24412741353	99567122655	331146179364	910059481422	2087007276627	4022124746409	6545498596110	9019473078408	10532600482794	10412080143894	8686112671808	6081573529460	3543856290546	1698021357318	657477286908	200602790856	46424563900	7658556100	802332000	40116600	64912164888576

b-files: A052141 (row sums), A316649 ?

Technical details

Node id: n-97a17a1dc7998ff8 · revision 0 · JSON
Definition source: oeis_catalog:A316649
Fingerprint: f6-bdbbd88be9205817

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A316649 triangle 64 terms, 8 rows · tabf · nonn · catalog b-file?

Row sums ? A052141

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 1,  2;
  2: 0, 1,  7,   12,     6;
  3: 0, 1, 14,   55,    92,     70,      20;
  4: 0, 1, 23,  153,   471,    780,     720,     350,       70;
  5: 0, 1, 34,  336,  1584,   4251,    7002,    7238,     4592,     1638,      252;
  6: 0, 1, 47,  640,  4210,  16175,   39733,   65226,    72660,    54390,    26250,     7392,      924;
  7: 0, 1, 62, 1107,  9596,  49225,  164898,  380731,   623576,   732618,   614700,   360162,   140184,    32604,     3432;
  8: 0, 1, 79, 1785, 19607, 128345,  555303, 1685257,  3716695,  6077196,  7430220,  6780312,  4557168,  2192190,   714714,   141570,   12870;
  9: 0, 1, 98, 2728, 36896, 298480, 1607648, 6139504, 17290304, 36808723, 60110030, 75805972, 73788176, 54980926, 30802772, 12563980, 3523520, 607750, 48620;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = (2*n^2-5*n+2+(6*n^2-15*n+6)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) - (n^2-3*n+2+(2*n^2-6*n+4)*x+(n^2-3*n+2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x + 2*x^2, P(2,x) = x + 7*x^2 + 12*x^3 + 6*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x + 2*x^2;
P[2, x_] := x + 7*x^2 + 12*x^3 + 6*x^4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*P[n - 1, x] - (n^2 - 3*n + 2 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x])/(n^2 - 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] = x + 2*x^2; v[3] = x + 7*x^2 + 12*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*v[i - 1] - (n^2 - 3*n + 2 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2])/(n^2 - 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

A316649 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1,  2;
  2: 0, 1,  7,   12,     6;
  3: 0, 1, 14,   55,    92,     70,      20;
  4: 0, 1, 23,  153,   471,    780,     720,     350,       70;
  5: 0, 1, 34,  336,  1584,   4251,    7002,    7238,     4592,     1638,      252;
  6: 0, 1, 47,  640,  4210,  16175,   39733,   65226,    72660,    54390,    26250,     7392,      924;
  7: 0, 1, 62, 1107,  9596,  49225,  164898,  380731,   623576,   732618,   614700,   360162,   140184,    32604,     3432;
  8: 0, 1, 79, 1785, 19607, 128345,  555303, 1685257,  3716695,  6077196,  7430220,  6780312,  4557168,  2192190,   714714,   141570,   12870;
  9: 0, 1, 98, 2728, 36896, 298480, 1607648, 6139504, 17290304, 36808723, 60110030, 75805972, 73788176, 54980926, 30802772, 12563980, 3523520, 607750, 48620;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = (2*n^2-5*n+2+(6*n^2-15*n+6)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) - (n^2-3*n+2+(2*n^2-6*n+4)*x+(n^2-3*n+2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x + 2*x^2, P(2,x) = x + 7*x^2 + 12*x^3 + 6*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x + 2*x^2;
P[2, x_] := x + 7*x^2 + 12*x^3 + 6*x^4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*P[n - 1, x] - (n^2 - 3*n + 2 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x])/(n^2 - 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] = x + 2*x^2; v[3] = x + 7*x^2 + 12*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*v[i - 1] - (n^2 - 3*n + 2 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2])/(n^2 - 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

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
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
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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

Used in 5

References

[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