A193722: Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.

A193722 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A193722

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	5	6						12
3	1	8	21	18					48
4	1	11	45	81	54				192
5	1	14	78	216	297	162			768
6	1	17	120	450	945	1053	486		3072
7	1	20	171	810	2295	3888	3645	1458	12288

All 15 rows b-files: A002001 (row sums), A164346 (row sums), A193722, A193723 (reversed triangle) ?

Technical details

Node id: n-6edd791871a0d803 · revision 1 · JSON
Definition source: oeis_catalog:A193722
Fingerprint: f6-6d5781c4134cee14

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): yes

Matches ?

A193723 reversed triangle 55 terms, 10 rows · tabl · nonn b-file?
A193722 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A110594, A002001, A164346, A092898

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  5,   6;
  3: 1,  8,  21,   18;
  4: 1, 11,  45,   81,   54;
  5: 1, 14,  78,  216,  297,   162;
  6: 1, 17, 120,  450,  945,  1053,   486;
  7: 1, 20, 171,  810, 2295,  3888,  3645,  1458;
  8: 1, 23, 231, 1323, 4725, 10773, 15309, 12393,  4374;
  9: 1, 26, 300, 2016, 8694, 24948, 47628, 58320, 41553, 13122;

%F Formula

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

%t Mathematica

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

A193722 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  5,   6;
  3: 1,  8,  21,   18;
  4: 1, 11,  45,   81,   54;
  5: 1, 14,  78,  216,  297,   162;
  6: 1, 17, 120,  450,  945,  1053,   486;
  7: 1, 20, 171,  810, 2295,  3888,  3645,  1458;
  8: 1, 23, 231, 1323, 4725, 10773, 15309, 12393,  4374;
  9: 1, 26, 300, 2016, 8694, 24948, 47628, 58320, 41553, 13122;

%F Formula

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

%t Mathematica

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

A193723 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:     1;
  1:     2,     1;
  2:     6,     5,     1;
  3:    18,    21,     8,     1;
  4:    54,    81,    45,    11,     1;
  5:   162,   297,   216,    78,    14,    1;
  6:   486,  1053,   945,   450,   120,   17,    1;
  7:  1458,  3645,  3888,  2295,   810,  171,   20,   1;
  8:  4374, 12393, 15309, 10773,  4725, 1323,  231,  23,  1;
  9: 13122, 41553, 58320, 47628, 24948, 8694, 2016, 300, 26, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (n-1+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (3*n - 3)*x)*P[n - 1, x])/(n - 1)]];
Table[Reverse[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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (3*n - 3)*x)*v[i - 1])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); 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(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), Liu–Wang theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers, Liu–Wang theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers, Liu–Wang theorem
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Iterated real-rootedness preservers

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
Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.