A123160: Triangle read by rows: T(n,k) = n!(n+k-1)!/((n-k)!(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.

A123160 ↗ nonnegative coefficients

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

Recurrence from A123160

\(\left(1 - n\right)\,P_{n}(x) = \left(6\,x - n - 4\,n x\right)\,P_{n-1}(x) + \left(5 + 14\,x - n - 4\,n x\right)\,P_{n-2}(x) + \left(-4 + n\right)\,P_{n-3}(x)\)

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

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

\(P_{3}(x) = 1 + 4\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	1	4	3													8
4	1	9	18	10												38
5	1	16	60	80	35											192
6	1	25	150	350	350	126										1002
7	1	36	315	1120	1890	1512	462									5336
8	1	49	588	2940	7350	9702	6468	1716								28814
9	1	64	1008	6720	23100	44352	48048	27456	6435							157184
10	1	81	1620	13860	62370	162162	252252	231660	115830	24310						864146
11	1	100	2475	26400	150150	504504	1051050	1372800	1093950	486200	92378					4780008
12	1	121	3630	47190	330330	1387386	3699696	6417840	7220070	5080790	2032316	352716				26572086
13	1	144	5148	80080	675675	3459456	11435424	25204608	37413090	36951200	23279256	8465184	1352078			148321344
14	1	169	7098	130130	1301300	7963956	31855824	86465808	162123390	210159950	184940756	105462084	35154028	5200300		830764794
15	1	196	9555	203840	2382380	17153136	81477396	266048640	611080470	995834840	1145210066	908596416	473227300	145608400	20058300	4666890936

b-files: A123160, A123164 (row sums) ?

Technical details

Node id: n-55697ddae4d22d60 · revision 0 · JSON
Definition source: oeis_catalog:A123160
Fingerprint: f6-82e5c201b4913233

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 -1)
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 ?

A123160 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A123164, A002003

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  4,    3;
   4: 1,  9,   18,    10;
   5: 1, 16,   60,    80,    35;
   6: 1, 25,  150,   350,   350,    126;
   7: 1, 36,  315,  1120,  1890,   1512,    462;
   8: 1, 49,  588,  2940,  7350,   9702,   6468,   1716;
   9: 1, 64, 1008,  6720, 23100,  44352,  48048,  27456,   6435;
  10: 1, 81, 1620, 13860, 62370, 162162, 252252, 231660, 115830, 24310;

%F Formula

The row polynomials satisfy -(n-1)*P(n,x) = -(n+(4*n-6)*x)*P(n-1,x) - (n-5+(4*n-14)*x)*P(n-2,x) + (n-4)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 4*x + 3*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 1 + 4*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (4*n - 6)*x)*P[n - 1, x] - (n - 5 + (4*n - 14)*x)*P[n - 2, x] + (n - 4)*P[n - 3, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 4*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(n + (4*n - 6)*x)*v[i - 1] - (n - 5 + (4*n - 14)*x)*v[i - 2] + (n - 4)*v[i - 3])/(-n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A123160 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,    3;
  3: 1,  9,   18,    10;
  4: 1, 16,   60,    80,    35;
  5: 1, 25,  150,   350,   350,    126;
  6: 1, 36,  315,  1120,  1890,   1512,    462;
  7: 1, 49,  588,  2940,  7350,   9702,   6468,   1716;
  8: 1, 64, 1008,  6720, 23100,  44352,  48048,  27456,   6435;
  9: 1, 81, 1620, 13860, 62370, 162162, 252252, 231660, 115830, 24310;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (4*n - 2)*x)*P[n - 1, x] - (n - 4 + (4*n - 10)*x)*P[n - 2, x] + (n - 3)*P[n - 3, x])/(-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] = 1 + x; v[3] = 1 + 4*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n + 1 + (4*n - 2)*x)*v[i - 1] - (n - 4 + (4*n - 10)*x)*v[i - 2] + (n - 3)*v[i - 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 ≥ 1, is zero or real-rooted with all roots in (−∞, 0)
by Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)

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

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.