veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	0	1															1
1	1	1															2
2	0	1	2	1													4
3	1	3	3	1													8
4	0	1	4	6	4	1											16
5	1	5	10	10	5	1											32
6	0	1	6	15	20	15	6	1									64
7	1	7	21	35	35	21	7	1									128
8	0	1	8	28	56	70	56	28	8	1							256
9	1	9	36	84	126	126	84	36	9	1							512
10	0	1	10	45	120	210	252	210	120	45	10	1					1024
11	1	11	55	165	330	462	462	330	165	55	11	1					2048
12	0	1	12	66	220	495	792	924	792	495	220	66	12	1			4096
13	1	13	78	286	715	1287	1716	1716	1287	715	286	78	13	1			8192
14	0	1	14	91	364	1001	2002	3003	3432	3003	2002	1001	364	91	14	1	16384

b-files: A000079 (row sums), A007318, A007318 (reversed triangle) ?

Technical details

Node id: n-6a690be6f27c7004 · revision 0 · JSON
Definition source: derived:veronese_parent_of:oeis_catalog:A109446
Fingerprint: f6-77a7ed3b1d5f257e

Observed on the stored rows ?

Degrees: 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15 (irregular)
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 ?

A007318 reversed triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?
A007318 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A000079, A166444, A034008, A011782, A131577, A220051, A118655, A050732

Submission text ?

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

%e Example

Triangle begins:
  0: 0, 1;
  1: 1, 1;
  2: 0, 1,  2,  1;
  3: 1, 3,  3,  1;
  4: 0, 1,  4,  6,   4,   1;
  5: 1, 5, 10, 10,   5,   1;
  6: 0, 1,  6, 15,  20,  15,  6,  1;
  7: 1, 7, 21, 35,  35,  21,  7,  1;
  8: 0, 1,  8, 28,  56,  70, 56, 28, 8, 1;
  9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;

%F Formula

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

%t Mathematica

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

A007318 triangle, offset 0, from P₀, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1, 1;
  2: 1, 2,  1;
  3: 1, 3,  3,  1;
  4: 1, 4,  6,  4,   1;
  5: 1, 5, 10, 10,   5,   1;
  6: 1, 6, 15, 20,  15,   6,  1;
  7: 1, 7, 21, 35,  35,  21,  7,  1;
  8: 1, 8, 28, 56,  70,  56, 28,  8, 1;
  9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := x;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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

A007318 reversed triangle, offset 0, from P₀, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1, 1;
  2: 1, 2,  1;
  3: 1, 3,  3,  1;
  4: 1, 4,  6,  4,   1;
  5: 1, 5, 10, 10,   5,   1;
  6: 1, 6, 15, 20,  15,   6,  1;
  7: 1, 7, 21, 35,  35,  21,  7,  1;
  8: 1, 8, 28, 56,  70,  56, 28,  8, 1;
  9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;

%F Formula

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

%t Mathematica

P[0, x_] := x;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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 Iterated real-rootedness preservers, Parent-row provers of polytool
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers, Parent-row provers of polytool
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Parent-row provers of polytool

Proof chains ? JSON

Parent-row provers of polytool polytool ? step › direct check
Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Used in 1

Binomial coefficients C(n,k) with n-k even, read by rows. Veronese coefficient section (r=2, j=1, shift 0)

Related ?

Same fingerprint ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.