A124860: A Jacobsthal-Pascal triangle.

A124860 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A124860

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	3	6	3						12
3	5	15	15	5					40
4	11	44	66	44	11				176
5	21	105	210	210	105	21			672
6	43	258	645	860	645	258	43		2752
7	85	595	1785	2975	2975	1785	595	85	10880

All 15 rows b-files: A110953 (row sums), A124860, A124860 (reversed triangle) ?

Technical details

Node id: n-9f2158ef8512cadd · revision 0 · JSON
Definition source: oeis_catalog:A124860
Fingerprint: f6-77a7ed3b1d5f257e

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

A124860 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A124860 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A110953, A003683

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   1,    1;
  2:   3,    6,     3;
  3:   5,   15,    15,     5;
  4:  11,   44,    66,    44,    11;
  5:  21,  105,   210,   210,   105,    21;
  6:  43,  258,   645,   860,   645,   258,    43;
  7:  85,  595,  1785,  2975,  2975,  1785,   595,    85;
  8: 171, 1368,  4788,  9576, 11970,  9576,  4788,  1368,  171;
  9: 341, 3069, 12276, 28644, 42966, 42966, 28644, 12276, 3069, 341;

%F Formula

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

%t Mathematica

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

A124860 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:   1;
  1:   1,    1;
  2:   3,    6,     3;
  3:   5,   15,    15,     5;
  4:  11,   44,    66,    44,    11;
  5:  21,  105,   210,   210,   105,    21;
  6:  43,  258,   645,   860,   645,   258,    43;
  7:  85,  595,  1785,  2975,  2975,  1785,   595,    85;
  8: 171, 1368,  4788,  9576, 11970,  9576,  4788,  1368,  171;
  9: 341, 3069, 12276, 28644, 42966, 42966, 28644, 12276, 3069, 341;

%F Formula

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

%t Mathematica

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

A124860 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:   1;
  1:   1,    1;
  2:   3,    6,     3;
  3:   5,   15,    15,     5;
  4:  11,   44,    66,    44,    11;
  5:  21,  105,   210,   210,   105,    21;
  6:  43,  258,   645,   860,   645,   258,    43;
  7:  85,  595,  1785,  2975,  2975,  1785,   595,    85;
  8: 171, 1368,  4788,  9576, 11970,  9576,  4788,  1368,  171;
  9: 341, 3069, 12276, 28644, 42966, 42966, 28644, 12276, 3069, 341;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

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

Related ?

Same fingerprint ?

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.