A120986: Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).

A120986 ↗ nonnegative coefficients

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

Recurrence from A120986

\(\left(n + 2\,n^{2}\right)\,P_{n}(x) = \left(1 + 2\,x - \frac{11}{2}\,n - 8\,n x + \frac{15}{2}\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-\frac{1}{2} - \frac{1}{2}\,x + x^{2} + \frac{1}{2}\,n + \frac{7}{2}\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	5	6	1						12
4	14	28	12	1					55
5	42	120	90	20	1				273
6	132	495	550	220	30	1			1428
7	429	2002	3003	1820	455	42	1		7752
8	1430	8008	15288	12740	4900	840	56	1	43263

All 15 rows b-files: A108767 (reversed triangle), A120986 ?

Technical details

Node id: n-4479b0484a11b442 · revision 0 · JSON
Definition source: oeis_catalog:A120986
Fingerprint: f6-85c07d3149a3188a

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 ?

A108767 reversed triangle 52 terms, 9 rows · tabl · nonn b-file?
A120986 triangle 50 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A001764

Submission text ?

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

%e Example

Triangle begins:
   1:     1;
   2:     2,      1;
   3:     5,      6,      1;
   4:    14,     28,     12,      1;
   5:    42,    120,     90,     20,      1;
   6:   132,    495,    550,    220,     30,      1;
   7:   429,   2002,   3003,   1820,    455,     42,     1;
   8:  1430,   8008,  15288,  12740,   4900,    840,    56,    1;
   9:  4862,  31824,  74256,  79968,  42840,  11424,  1428,   72,  1;
  10: 16796, 125970, 348840, 465120, 325584, 122094, 23940, 2280, 90, 1;

%F Formula

The row polynomials satisfy (2*n^2+n)*P(n,x) = ((15/2)*n^2-(11/2)*n+1+(6*n^2-8*n+2)*x)*P(n-1,x) + ((1/2)*n-1/2+((7/2)*n-1/2)*x-(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((15/2)*n^2 - (11/2)*n + 1 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((1/2)*n - 1/2 + ((7/2)*n - 1/2)*x - (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((15/2)*n^2 - (11/2)*n + 1 + (6*n^2 - 8*n + 2)*x)*v[i - 1] + ((1/2)*n - 1/2 + ((7/2)*n - 1/2)*x - (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); 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

A108767 reversed triangle, offset 1, from P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,    5;
   4: 1, 12,   28,    14;
   5: 1, 20,   90,   120,     42;
   6: 1, 30,  220,   550,    495,    132;
   7: 1, 42,  455,  1820,   3003,   2002,    429;
   8: 1, 56,  840,  4900,  12740,  15288,   8008,   1430;
   9: 1, 72, 1428, 11424,  42840,  79968,  74256,  31824,   4862;
  10: 1, 90, 2280, 23940, 122094, 325584, 465120, 348840, 125970, 16796;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n^2+n)*P(n,x) = ((15/2)*n^2-(11/2)*n+1+(6*n^2-8*n+2)*x)*P(n-1,x) + ((1/2)*n-1/2+((7/2)*n-1/2)*x-(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((15/2)*n^2 - (11/2)*n + 1 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((1/2)*n - 1/2 + ((7/2)*n - 1/2)*x - (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((15/2)*n^2 - (11/2)*n + 1 + (6*n^2 - 8*n + 2)*x)*v[i - 1] + ((1/2)*n - 1/2 + ((7/2)*n - 1/2)*x - (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A120986 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:     1;
  1:     2,      1;
  2:     5,      6,      1;
  3:    14,     28,     12,      1;
  4:    42,    120,     90,     20,      1;
  5:   132,    495,    550,    220,     30,      1;
  6:   429,   2002,   3003,   1820,    455,     42,     1;
  7:  1430,   8008,  15288,  12740,   4900,    840,    56,    1;
  8:  4862,  31824,  74256,  79968,  42840,  11424,  1428,   72,  1;
  9: 16796, 125970, 348840, 465120, 325584, 122094, 23940, 2280, 90, 1;

%F Formula

The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = ((15/2)*n^2+(19/2)*n+3+(6*n^2+4*n)*x)*P(n-1,x) + ((1/2)*n+((7/2)*n+3)*x-(4*n+3)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((15/2)*n^2 + (19/2)*n + 3 + (6*n^2 + 4*n)*x)*P[n - 1, x] + ((1/2)*n + ((7/2)*n + 3)*x - (4*n + 3)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*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; for(n = 1, N, my(i = n + 1); v[i] = (((15/2)*n^2 + (19/2)*n + 3 + (6*n^2 + 4*n)*x)*v[i - 1] + ((1/2)*n + ((7/2)*n + 3)*x - (4*n + 3)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 5*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 ≥ 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.