A119308: Triangle for first differences of Catalan numbers.

A119308 ↗ nonnegative coefficients

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

Recurrence from A119308

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

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

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

\(P_{2}(x) = 3 + 5\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	3	5	1						9
3	4	14	9	1					28
4	5	30	40	14	1				90
5	6	55	125	90	20	1			297
6	7	91	315	385	175	27	1		1001
7	8	140	686	1274	980	308	35	1	3432

All 15 rows b-files: A119308 ?

Technical details

Node id: n-a88a492d75692c65 · revision 0 · JSON
Definition source: oeis_catalog:A119308
Fingerprint: f6-1c718f648220d69b

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 ?

A119308 triangle 58 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A000245, A071724

Submission text ?

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

%e Example

Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   5,    1;
  3:  4,  14,    9,    1;
  4:  5,  30,   40,   14,     1;
  5:  6,  55,  125,   90,    20,     1;
  6:  7,  91,  315,  385,   175,    27,    1;
  7:  8, 140,  686, 1274,   980,   308,   35,   1;
  8:  9, 204, 1344, 3528,  4116,  2184,  504,  44,  1;
  9: 10, 285, 2430, 8568, 14112, 11340, 4410, 780, 54, 1;

%F Formula

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

%t Mathematica

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

A119308 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   5,    1;
  3:  4,  14,    9,    1;
  4:  5,  30,   40,   14,     1;
  5:  6,  55,  125,   90,    20,     1;
  6:  7,  91,  315,  385,   175,    27,    1;
  7:  8, 140,  686, 1274,   980,   308,   35,   1;
  8:  9, 204, 1344, 3528,  4116,  2184,  504,  44,  1;
  9: 10, 285, 2430, 8568, 14112, 11340, 4410, 780, 54, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 3 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 4 + (2*n + 3)*x)*P[n - 1, x] - (3*n - 1 - x + n*x^2)*P[n - 2, x] + (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*P[n - 3, x])/(n + 3)]];
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; v[3] = 3 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n + 4 + (2*n + 3)*x)*v[i - 1] - (3*n - 1 - x + n*x^2)*v[i - 2] + (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*v[i - 3])/(n + 3)); 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)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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

Used in 11

3 more

Related ?

Same fingerprint ?

A119308 → reverse

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.