A109446: Binomial coefficients C(n,k) with n-k even, read by rows.

Veronese parent of A109446 → Veronese 2 (j=1)

A109446 ↗ nonnegative coefficients

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

Recurrence from A109446

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

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

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

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

\(P_{3}(x) = 3 + x\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	1	1			2
3	3	1			4
4	1	6	1		8
5	5	10	1		16
6	1	15	15	1	32
7	7	35	21	1	64

All 15 rows b-files: A011782 (row sums), A034839 (reversed triangle), A109446 ?

Technical details

Node id: n-a5420872eb2b873d · revision 0 · JSON
Definition source: oeis_catalog:A109446
Fingerprint: f6-4e15cddc85973b47

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 ?

A109446 triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?
A034839 reversed triangle 56 terms, 14 rows · tabf · nonn · catalog b-file?

Row sums ? A166444, A274865, A274864, A274866, A220493, A011782, A220469, A249169

Submission text ?

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

%e Example

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

%F Formula

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

%t Mathematica

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

A034839 reversed triangle, offset 0, from P₀

%e Example

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

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-2,x) - (1-2*x+x^2)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 3 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] - (1 - 2*x + x^2)*P[n - 4, x]];
Table[Reverse[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; v[3] = 1 + x; v[4] = 3 + x; for(n = 4, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] - (1 - 2*x + x^2)*v[i - 4]); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A109446 triangle, offset 0, from P₀

%e Example

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

%F Formula

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

%t Mathematica

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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Veronese sections of Pólya frequency sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Veronese sections of Pólya frequency sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Veronese sections of Pólya frequency sequences

Proof chains ? JSON

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
- from veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

References

[ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[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
[Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.